Explanation:
Okay, so I need to figure out how many different values can be obtained from the expression 2·3·4·5 + 1 by inserting parentheses. The original value is 121 because multiplication is done before addition. But if we can change the order of operations by adding parentheses, we might get different results. The example given is (2·(3·4))·(5+1) = 144. So, my task is to find all possible values by inserting parentheses in different ways and count them.

First, let me recall that inserting parentheses can change the order in which operations are performed. Since there are both multiplications and an addition, I need to consider how parentheses can affect the order of these operations.

The original expression is 2·3·4·5 + 1. Let me note that the operations are all multiplications except for the addition at the end. So, the expression is a sequence of multiplications followed by an addition. However, inserting parentheses can group some of the multiplications together or even group the addition with some multiplications.

But wait, the addition is only between the product of 2·3·4·5 and 1. However, if we insert parentheses, we can change when the addition happens. For example, if we group the addition with some of the multiplications, we might get different results.

Let me think about the structure of the expression. The original expression is:

( ( (2·3)·4 )·5 ) + 1 = 121

But if we insert parentheses differently, we can change the order. However, since all the operations are multiplications except for the addition, the addition is only between the entire product of 2·3·4·5 and 1. But if we can insert parentheses to make the addition happen earlier, that would change the value.

Wait, but the original expression is written as 2·3·4·5 + 1. So, according to order of operations, multiplication is done before addition. However, if we insert parentheses, we can make the addition happen before some multiplications. For example, if we do 2·(3·4·5 + 1), that would be different.

So, the key is that the addition is between the product of 2·3·4·5 and 1, but if we can parenthesize to make the addition occur at different points.

But how many different ways can we insert parentheses here? Let me think about the expression as a sequence of operations. Let me consider the expression as a sequence of numbers and operations: 2, ·, 3, ·, 4, ·, 5, +, 1.

So, there are four multiplications and one addition. The question is about inserting parentheses to change the order of operations. However, since multiplication is associative and commutative, but addition is not. However, since we are only allowed to insert parentheses, not rearrange terms, the positions of the numbers are fixed.

So, the problem reduces to determining all possible values that can be obtained by parenthesizing the expression 2·3·4·5 + 1.

Let me think about the different ways to parenthesize. Since there are multiple multiplications and one addition, the addition can be grouped with some of the multiplications.

First, let me note that the addition is between the product of 2·3·4·5 and 1. However, if we can insert parentheses to make the addition occur before some multiplications. For example, if we do 2·(3·(4·(5 + 1))), which would be 2·(3·(4·6)) = 2·(3·24) = 2·72 = 144. Wait, that's the same as the example given.

Alternatively, if we do (2·3)·(4·(5 + 1)) = 6·(4·6) = 6·24 = 144. Same result.

Alternatively, if we do ((2·3)·4)·(5 + 1) = 24·6 = 144. Also same.

So, different parenthesizations can lead to the same value. So, the key is to find all unique values that can be obtained through different parenthesizations.

So, perhaps I need to consider all possible ways to parenthesize the expression, compute the value for each, and then count the unique ones.

But how do I systematically approach this?

First, let me note that the expression has five numbers and four operations: 2, ·, 3, ·, 4, ·, 5, +, 1.

But the addition is only between the product of the first four numbers and 1. However, inserting parentheses can change when the addition is performed.

Alternatively, maybe I can think of the expression as two parts: the product of 2, 3, 4, 5 and the addition with 1. But if we can insert parentheses to make the addition happen earlier.

But since the addition is only between the product and 1, but if we can group some of the multiplications with the addition.

Wait, for example, if we do 2·(3·(4·(5 + 1))) as above. Or (2·(3·(4·5))) + 1, which is the original expression.

Alternatively, maybe grouping the addition with some of the numbers. For example, (2·3·4·(5 + 1)) which is 2·3·4·6 = 144.

Alternatively, ((2·3)·(4·(5 + 1))) = 6·(4·6) = 6·24 = 144.

Alternatively, (2·(3·4·(5 + 1))) = 2·(3·4·6) = 2·72 = 144.

Alternatively, ((2·(3·4))·(5 + 1)) = (2·12)·6 = 24·6 = 144.

So, all these different parenthesizations lead to 144.

But what if we do something else? For example, if we do 2·(3·(4·5 + 1))? Let me compute that.

First, compute 4·5 + 1 = 20 + 1 = 21. Then 3·21 = 63. Then 2·63 = 126.

So that's a different value.

Similarly, if we do (2·3)·(4·5 + 1) = 6·(20 + 1) = 6·21 = 126.

Same result.

Alternatively, 2·(3·4·5 + 1) = 2·(60 + 1) = 2·61 = 122.

Wait, that's different.

So, that's another value.

So, here we have different values: 121, 122, 126, 144.

Wait, but maybe there are more. Let me check.

Let me think of all possible ways to insert parentheses.

First, the original expression is ((2·3)·4)·5 + 1 = 121.

But if we can insert parentheses to make the addition happen at different points.

So, the addition is between the product of the first four numbers and 1. But if we can insert parentheses to make the addition happen earlier.

So, the addition can be grouped with some of the multiplications.

Let me think of the expression as a combination of multiplications and an addition.

Let me consider the addition as an operation that can be applied to different parts.

So, the expression is: [expression] + 1, where [expression] is 2·3·4·5. But if we can parenthesize to have the addition occur earlier.

Alternatively, maybe the addition is between some of the multiplications and 1.

So, for example, if we have 2·(3·(4·(5 + 1))) as before.

Or, (2·(3·4))·(5 + 1).

Or, 2·( (3·4)·(5 + 1) ).

Or, (2·3)·(4·(5 + 1)).

Or, ((2·3)·4)·(5 + 1).

Or, 2·(3·(4·5 + 1)).

Or, (2·(3·(4·5))) + 1.

Wait, but that's the original expression.

So, perhaps there are different ways to parenthesize the addition.

Let me think of the addition as an operation that can be inserted between different parts.

So, the original expression is ( ( (2·3)·4 )·5 ) + 1.

But if we can insert parentheses to make the addition happen between different terms.

So, for example, if we do 2·3·(4·(5 + 1)) which is 2·3·(4·6) = 2·3·24 = 144.

Or, 2·(3·4·(5 + 1)) = 2·(3·4·6) = 2·72 = 144.

Or, (2·3·4)·(5 + 1) = 24·6 = 144.

Or, 2·( (3·4)·(5 + 1) ) = same as above.

Alternatively, if we do (2·(3·(4·5))) + 1 = 2·(3·20) + 1 = 2·60 + 1 = 121.

Alternatively, if we do 2·(3·(4·5 + 1)) = 2·(3·(20 + 1)) = 2·(3·21) = 2·63 = 126.

Alternatively, (2·3)·(4·(5 + 1)) = 6·(4·6) = 6·24 = 144.

Alternatively, (2·(3·4))·(5 + 1) = (2·12)·6 = 24·6 = 144.

Alternatively, ((2·3)·4)·(5 + 1) = 24·6 = 144.

Alternatively, 2·(3·(4·(5 + 1))) = 2·(3·(4·6)) = 2·(3·24) = 2·72 = 144.

Alternatively, 2·( (3·4)·(5 + 1) ) = same as above.

Alternatively, ( (2·(3·4))·5 ) + 1 = (2·12·5) + 1 = 120 + 1 = 121.

Wait, that's the original expression.

Alternatively, if we do (2·(3·(4·5))) + 1 = same as original.

Alternatively, if we do ( (2·3)·(4·5) ) + 1 = same as original.

Alternatively, if we do 2·(3·4·5 + 1) = 2·(60 + 1) = 2·61 = 122.

So, that's another value.

So, so far, I have found values: 121, 122, 126, 144.

Are there more? Let me check.

What if we do something like (2·3·(4·5)) + 1 = same as original.

Alternatively, 2·(3·(4·5 + 1)) = 126.

Alternatively, 2·(3·4·(5 + 1)) = 144.

Alternatively, 2·( (3·4)·(5 + 1) ) = same.

Alternatively, (2·(3·4))·(5 + 1) = 144.

Alternatively, (2·(3·(4·5))) + 1 = 121.

Alternatively, ( (2·3)·(4·(5 + 1)) ) = 144.

Alternatively, ( (2·(3·4))·(5 + 1) ) = 144.

Alternatively, ( ( (2·3)·4 )·(5 + 1) ) = 144.

Alternatively, 2·( (3·(4·5)) + 1 ) = 2·(60 + 1) = 122.

Wait, that's the same as before.

Alternatively, if we do (2·3)·(4·5 + 1) = 6·(20 + 1) = 6·21 = 126.

So, same as before.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do 2·( (3·4)·(5 + 1) ) = 144.

Is there a way to get another value? Let me think.

What if we do something like (2·(3·(4 + 5)))·1? Wait, but the original expression is 2·3·4·5 + 1. So, the numbers are fixed. The addition is only between the product and 1. Wait, but if we can insert parentheses to make the addition happen between different numbers.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product of 2·3·4·5 and 1. However, if we can insert parentheses to make the addition happen between some of the multiplications and 1.

Wait, for example, if we do 2·(3·4·5 + 1). That is, adding 1 to the product of 3·4·5 first, then multiplying by 2.

Similarly, if we do (2·3·4)·(5 + 1) = 24·6 = 144.

Alternatively, if we do (2·3)·(4·(5 + 1)) = 6·24 = 144.

Alternatively, if we do 2·( (3·4)·(5 + 1) ) = same.

Alternatively, if we do 2·(3·(4·(5 + 1))) = same.

Alternatively, if we do (2·(3·(4·5))) + 1 = 121.

Alternatively, if we do ( (2·3)·4 )·(5 + 1) = 24·6 = 144.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do (2·(3·4·5)) + 1 = 121.

Alternatively, if we do 2·(3·4·(5 + 1)) = 144.

Alternatively, if we do (2·3)·(4·5 + 1) = 6·21 = 126.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do (2·(3·(4·5))) + 1 = 121.

Alternatively, if we do (2·(3·4))·(5 + 1) = 24·6 = 144.

So, so far, the values I have are 121, 122, 126, 144.

Wait, where does 122 come from? That was when we did 2·(3·4·5 + 1) = 2·(60 + 1) = 122.

Is there another way to get a different value? Let me think.

What if we do something like (2·3·(4·5)) + 1 = same as original.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do (2·(3·(4 + 5)))·1? Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. If we try to add 4 and 5 first, that would be changing the original expression. Wait, but the original expression is 2·3·4·5 + 1. So, the numbers are fixed. So, inserting parentheses can't change the order of the numbers.

Wait, but if we do 2·(3·(4 + 5))·1? Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. So, inserting parentheses can't change the positions of the numbers.

Wait, but if we do 2·(3·(4 + 5)) + 1? Wait, that would be different. Let me check.

But the original expression is 2·3·4·5 + 1. If we insert parentheses to make it 2·(3·(4 + 5)) + 1, that would be allowed? Wait, but the original expression is written as 2·3·4·5 + 1. So, inserting parentheses can only group operations, not change the order of numbers.

Wait, but if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. But in the original expression, the addition is only between the product and 1. However, inserting parentheses can allow the addition to happen between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product of 2·3·4·5 and 1. However, inserting parentheses can change the order of operations. For example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, if we do 2·(3·(4·(5 + 1))), that is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. However, in the original expression, there is no addition between 4 and 5. Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is written as 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition. So, the addition is only between the product of the first four numbers and 1. However, inserting parentheses can change the order of operations. For example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, if we do 2·(3·(4·(5 + 1))), that is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. However, in the original expression, there is no addition between 4 and 5. Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product of 2·3·4·5 and 1. However, if we insert parentheses to make the addition happen between 4 and 5, that would be changing the original expression. Wait, but the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Let me think. If we have 2·3·4·5 + 1, inserting parentheses can change the order. For example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, if we do 2·(3·(4·(5 + 1))). But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. However, in the original expression, there is no addition between 4 and 5. Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Because the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product of 2·3·4·5 and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Because the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Because the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, maybe I need to think of the expression as having operations between numbers. Let me write down the expression with all operations:

2 * 3 * 4 * 5 + 1

So, the operations are:

Between 2 and 3: multiplication

Between 3 and 4: multiplication

Between 4 and 5: multiplication

Between 5 and 1: addition

So, the addition is only between 5 and 1. Therefore, inserting parentheses can only group the multiplications and the addition.

Therefore, the addition is only between 5 and 1. Therefore, inserting parentheses can't change the fact that the addition is between 5 and 1. Wait, but that seems contradictory to the example given.

Wait, the example given is (2·(3·4))·(5 + 1) = 144. So, in this case, the addition is between 5 and 1, but they are grouped with the multiplication.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is between the product of 2·3·4·5 and 1. However, inserting parentheses can group the addition with some of the multiplications.

Wait, but the addition is only between 5 and 1. So, if we want to group the addition with some multiplications, we need to have parentheses around 5 + 1, and then multiply that with other terms.

So, for example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, (2·3·4)·(5 + 1).

But if we want to do something like 2·(3·(4 + 5)) + 1, that would require changing the order of operations, but in the original expression, the addition is only between 5 and 1. Therefore, inserting parentheses can't change the fact that the addition is between 5 and 1. Wait, but how is the example given possible?

Wait, the example is (2·(3·4))·(5 + 1). So, that is grouping the multiplication of 3 and 4, then multiplying by 2, then multiplying by (5 + 1). So, the addition is still between 5 and 1.

Therefore, the addition is always between 5 and 1. Therefore, inserting parentheses can only group the multiplications and the addition.

Therefore, the addition is fixed between 5 and 1. Therefore, the only way to change the value is by grouping the multiplications in different ways, and then combining with the addition.

Wait, but then how can we get 126? For example, 2·(3·(4·5 + 1)) = 2·(3·(20 + 1)) = 2·(3·21) = 126.

But in this case, the addition is between 5 and 1, but we have 4·5 + 1. Wait, but that would require that the addition is between 5 and 1, but also between 4 and 5?

Wait, no. Wait, if we have 4·5 + 1, that would be 4·5 + 1, which is 20 + 1 = 21. But in the original expression, the addition is only between 5 and 1. Therefore, inserting parentheses to do 4·(5 + 1) is allowed, because that is grouping the addition between 5 and 1 with the multiplication by 4.

Wait, so the addition is between 5 and 1, but we can group that addition with previous multiplications.

Therefore, the addition is fixed between 5 and 1, but we can group that addition with previous multiplications.

Therefore, the different values come from different ways of grouping the multiplications before the addition.

So, the expression can be thought of as (some product of 2, 3, 4, 5) + 1, where the product can be grouped in different ways.

Wait, but if we do 2·(3·(4·(5 + 1))), that is grouping the addition with 5 and 1, then multiplying by 4, then by 3, then by 2.

Alternatively, if we do (2·3·4)·(5 + 1), that is grouping the addition with 5 and 1, then multiplying by 2·3·4.

Alternatively, if we do 2·(3·(4·5 + 1)), that is grouping the addition with 5 and 1, but also multiplying by 4? Wait, no. Wait, 4·5 + 1 is 21, then multiplied by 3, then multiplied by 2.

Wait, but in this case, the addition is between 5 and 1, but we have 4·(5 + 1) or (4·5) + 1?

Wait, if we have 4·(5 + 1), that is grouping the addition with 5 and 1, then multiplying by 4.

If we have (4·5) + 1, that is multiplying 4 and 5 first, then adding 1.

So, depending on how we group the operations, we can have different results.

Therefore, the key is that the addition is between 5 and 1, but we can choose to group that addition with previous multiplications or not.

Therefore, the different values come from different ways of grouping the multiplications before the addition.

So, the problem reduces to: how many different values can be obtained by grouping the multiplications of 2, 3, 4, 5 in different ways, and then adding 1.

Wait, but the addition is only between the product and 1. However, if we group the addition with some multiplications, we can have different results.

Wait, but if we think of the entire expression as (some expression involving 2, 3, 4, 5) + 1, where the "some expression" can be grouped in different ways.

Therefore, the number of different values depends on the number of different ways to parenthesize the multiplications of 2, 3, 4, 5.

But since multiplication is associative, different parenthesizations of the same product give the same result. Wait, but if we have different groupings, but since multiplication is associative, they all give the same result.

Wait, but in our previous examples, we saw that different parenthesizations can lead to different results when combined with the addition.

Wait, for example, if we have (2·3·4·5) + 1 = 120 + 1 = 121.

But if we have 2·(3·(4·(5 + 1))) = 2·(3·(4·6)) = 2·(3·24) = 144.

But here, the addition is between 5 and 1, but we have grouped the multiplication with 4 and the addition.

So, in this case, the product is not the entire product of 2, 3, 4, 5, but rather, some of the multiplications are combined with the addition.

Therefore, the key is that the addition can be combined with some of the multiplications.

Therefore, the problem is not just about parenthesizing the multiplications, but also about when the addition is applied.

Therefore, the addition can be applied at different points.

So, the addition is between 5 and 1, but we can choose to apply it before multiplying by 4, or before multiplying by 3, etc.

Therefore, the different values come from different ways of inserting the addition into the multiplication sequence.

So, for example, if we consider the expression as a sequence of operations: 2 * 3 * 4 * 5 + 1.

We can think of the addition as an operation that can be inserted at different positions.

But since the addition is only between 5 and 1, we can think of inserting parentheses to group the addition with some of the multiplications.

So, the addition can be grouped with the last multiplication (i.e., 5 + 1), or with the previous multiplication (i.e., 4*(5 + 1)), or with the previous previous multiplication (i.e., 3*(4*(5 + 1))), etc.

Therefore, the number of different values depends on how many different ways we can insert the addition into the multiplication sequence.

So, for example, the addition can be inserted after the first multiplication (i.e., 2*(3*(4*(5 + 1)))), after the second multiplication (i.e., (2*3)*(4*(5 + 1))), after the third multiplication (i.e., ((2*3)*4)*(5 + 1)), or after the fourth multiplication (i.e., (((2*3)*4)*5) + 1).

Wait, but inserting the addition after the fourth multiplication is the original expression.

But inserting the addition after the third multiplication would be ((2*3)*4)*(5 + 1) = 24*6 = 144.

Inserting the addition after the second multiplication would be (2*3)*(4*(5 + 1)) = 6*24 = 144.

Inserting the addition after the first multiplication would be 2*(3*(4*(5 + 1))) = 2*(3*24) = 144.

So, all these give the same value.

But earlier, we saw that if we do 2*(3*(4*5 + 1)) = 2*(3*21) = 126.

Wait, how is that different?

Ah, here, the addition is between 5 and 1, but we have 4*5 + 1 instead of 4*(5 + 1).

So, this is different.

Therefore, the addition can be applied at different points.

So, the addition can be applied either before or after some multiplications.

Wait, but how?

Wait, the original expression is 2·3·4·5 + 1.

If we want to apply the addition before multiplying by 4, we need to have 4*(5 + 1).

If we want to apply the addition before multiplying by 3, we need to have 3*(4*(5 + 1)).

If we want to apply the addition before multiplying by 2, we need to have 2*(3*(4*(5 + 1))).

Alternatively, if we apply the addition after multiplying by 4, we have (4*5) + 1.

Similarly, applying the addition after multiplying by 3: (3*4*5) + 1.

Wait, but how does that work?

Wait, if we have 2*(3*(4*5) + 1).

So, that would be 2*( (3*(4*5)) + 1 ) = 2*(60 + 1) = 122.

Similarly, if we have (2*3)*(4*5 + 1) = 6*(20 + 1) = 6*21 = 126.

So, here, the addition is applied after multiplying by 4, but before multiplying by 3 or 2.

Therefore, the addition can be applied at different points in the multiplication sequence.

Therefore, the different values come from different positions where the addition is applied.

Therefore, the problem reduces to determining how many different positions we can apply the addition, and what values that gives.

So, the addition is between 5 and 1.

But we can choose to apply the addition at different points.

Let me think of the expression as follows:

We have the numbers 2, 3, 4, 5, and then +1.

The operations between the numbers are all multiplications, except for the last operation which is addition.

But inserting parentheses can change when the addition is applied.

So, the addition can be applied after multiplying 2, 3, 4, or 5.

Wait, but how?

Let me think of the expression as a sequence of operations.

Let me consider the expression as follows:

Start with 2, then multiply by 3, then multiply by 4, then multiply by 5, then add 1.

But inserting parentheses can change the order.

For example, if we add parentheses to apply the addition before multiplying by 4, we get:

2 * 3 * (4 * (5 + 1))

Which is 2 * 3 * (4 * 6) = 2 * 3 * 24 = 144.

Alternatively, if we apply the addition before multiplying by 3:

2 * (3 * (4 * (5 + 1))) = same as above.

Alternatively, if we apply the addition before multiplying by 2:

2 * (3 * (4 * (5 + 1))) = same.

Alternatively, if we apply the addition after multiplying by 5, which is the original expression:

(2 * 3 * 4 * 5) + 1 = 121.

Alternatively, if we apply the addition after multiplying by 4:

(2 * 3 * 4) * (5 + 1) = 24 * 6 = 144.

Alternatively, if we apply the addition after multiplying by 3:

(2 * 3) * (4 * (5 + 1)) = 6 * 24 = 144.

Alternatively, if we apply the addition after multiplying by 2:

2 * (3 * (4 * (5 + 1))) = 144.

Alternatively, if we apply the addition between multiplying by 4 and 5, but that would be (4 * 5) + 1, which is 21. Then, multiplying by 3 and 2.

So, for example:

2 * (3 * ( (4 * 5) + 1 )) = 2 * (3 * 21) = 126.

Similarly, (2 * 3) * ( (4 * 5) + 1 ) = 6 * 21 = 126.

Alternatively, 2 * 3 * ( (4 * 5) + 1 ) = 6 * 21 = 126.

Alternatively, if we apply the addition after multiplying by 5, but that's the original expression.

Alternatively, if we apply the addition after multiplying by 3 and 4, but that would be ( (2 * 3) * 4 ) * (5 + 1) = 24 * 6 = 144.

Alternatively, if we apply the addition after multiplying by 2, 3, and 4, which is the original expression.

Alternatively, if we apply the addition after multiplying by 2 and 3, which is ( (2 * 3) * 4 * 5 ) + 1 = 120 + 1 = 121.

Alternatively, if we apply the addition after multiplying by 2, which would be 2 * ( (3 * 4 * 5) + 1 ) = 2 * (60 + 1) = 122.

So, here, applying the addition after multiplying by 2, but before multiplying by 3, 4, 5.

Wait, but how is that possible?

If we have 2 * ( (3 * 4 * 5) + 1 ), that would mean that we first compute 3 * 4 * 5, then add 1, then multiply by 2.

But in the original expression, the addition is between 5 and 1. So, inserting parentheses to do 3 * 4 * 5 + 1 is allowed?

Wait, but the original expression is 2 * 3 * 4 * 5 + 1. So, the addition is only between 5 and 1. However, inserting parentheses to do 3 * 4 * 5 + 1 is allowed, because that is grouping the addition with the previous multiplications.

Therefore, the addition can be applied at different points in the multiplication sequence.

So, the different values come from different positions where the addition is applied.

So, the addition can be applied after multiplying by 2, 3, 4, or 5.

Wait, but how?

Let me think of the expression as follows:

The addition is between 5 and 1.

But we can choose to apply the addition at different points by grouping.

For example:

Case 1: Apply the addition after multiplying by 5: (2 * 3 * 4 * 5) + 1 = 121.

Case 2: Apply the addition after multiplying by 4: (2 * 3 * 4) * (5 + 1) = 24 * 6 = 144.

Case 3: Apply the addition after multiplying by 3: (2 * 3) * (4 * (5 + 1)) = 6 * 24 = 144.

Case 4: Apply the addition after multiplying by 2: 2 * (3 * (4 * (5 + 1))) = 2 * (3 * 24) = 144.

Case 5: Apply the addition between multiplying by 4 and 5: 2 * 3 * ( (4 * 5) + 1 ) = 6 * 21 = 126.

Case 6: Apply the addition between multiplying by 3 and 4: 2 * (3 * (4 * 5) + 1 ) = 2 * (60 + 1) = 122.

Case 7: Apply the addition between multiplying by 2 and 3: (2 * (3 * 4 * 5)) + 1 = 120 + 1 = 121.

Wait, but case 7 is same as case 1.

Similarly, case 5 and case 6 are different.

So, so far, we have values: 121, 122, 126, 144.

Are there more cases?

Wait, can we apply the addition in more positions?

For example, if we apply the addition between multiplying by 2 and 3, but also between multiplying by 4 and 5?

Wait, but the addition is only between 5 and 1.

Wait, but if we have something like 2 * ( (3 * 4) * (5 + 1) ), which is case 3.

Alternatively, if we have (2 * (3 * (4 * (5 + 1)))) which is case 4.

Alternatively, if we have ( (2 * 3) * (4 * (5 + 1)) ) which is case 3.

Alternatively, if we have (2 * 3) * (4 * (5 + 1)) = same as case 3.

Alternatively, if we have 2 * (3 * (4 * (5 + 1))) = case 4.

Alternatively, if we have 2 * ( (3 * 4) * (5 + 1) ) = same as case 3.

So, all these give 144.

Alternatively, if we have 2 * (3 * ( (4 * 5) + 1 )) = case 6: 122.

Alternatively, if we have (2 * 3) * ( (4 * 5) + 1 ) = case 5: 126.

Alternatively, if we have 2 * 3 * ( (4 * 5) + 1 ) = same as case 5: 126.

Alternatively, if we have (2 * (3 * 4 * 5)) + 1 = case 1: 121.

Alternatively, if we have ( (2 * 3 * 4) * 5 ) + 1 = case 1: 121.

Alternatively, if we have ( (2 * 3) * 4 * 5 ) + 1 = case 1: 121.

So, seems like the only different values are 121, 122, 126, 144.

But let me check if there are other possibilities.

For example, can we apply the addition in more than one place?

Wait, the addition is only between 5 and 1. So, we can't have multiple additions.

But if we have something like 2 * (3 + (4 * 5)) + 1? Wait, but that would require changing the original expression.

Wait, the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses can't introduce new additions.

Therefore, the only addition is between 5 and 1.

Therefore, the only way to apply the addition is either before multiplying by 4, 3, 2, or after multiplying by 5.

Wait, but how?

Wait, if we apply the addition after multiplying by 5, that's the original expression.

If we apply the addition before multiplying by 4, that is grouping 5 + 1 with 4.

If we apply the addition before multiplying by 3, that is grouping 5 + 1 with 4 and 3.

If we apply the addition before multiplying by 2, that is grouping 5 + 1 with 4, 3, and 2.

Alternatively, if we apply the addition between multiplying by 4 and 5, that is grouping 4 * 5 + 1.

Similarly, applying the addition between multiplying by 3 and 4, that is grouping 3 * (4 * 5) + 1.

Wait, but how is that different?

Wait, if we have 2 * (3 * (4 * 5) + 1) = 2 * (60 + 1) = 122.

Alternatively, if we have 2 * 3 * (4 * 5 + 1) = 6 * 21 = 126.

Alternatively, if we have (2 * 3) * (4 * 5 + 1) = 6 * 21 = 126.

Alternatively, if we have 2 * (3 * 4 * 5 + 1) = 2 * 61 = 122.

So, these are two different values: 122 and 126.

So, in total, we have:

- Applying the addition after multiplying by 5: 121.

- Applying the addition after multiplying by 4: 144.

- Applying the addition after multiplying by 3: 144.

- Applying the addition after multiplying by 2: 144.

- Applying the addition between multiplying by 4 and 5: 126.

- Applying the addition between multiplying by 3 and 4: 122.

Wait, but is there another way to apply the addition?

For example, applying the addition between multiplying by 2 and 3?

But that would be (2 * (3 * 4 * 5)) + 1 = 120 + 1 = 121, which is same as case 1.

Alternatively, applying the addition between multiplying by 2 and 3, but also between multiplying by 4 and 5?

But that would require multiple additions, which isn't allowed.

Therefore, the only different values are 121, 122, 126, 144.

Wait, but let me check if there are other possibilities.

For example, if we have something like (2 * (3 + 4)) * (5 + 1). Wait, but that would require changing the original expression.

Wait, the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses can't introduce new additions.

Therefore, the only addition is between 5 and 1.

Therefore, the only different values are 121, 122, 126, 144.

But let me check if there are more possibilities.

For example, if we have 2 * ( (3 * 4) * (5 + 1) ) = 144.

Or, if we have ( (2 * 3) * 4 ) * (5 + 1) = 144.

Or, if we have 2 * (3 * (4 * (5 + 1))) = 144.

So, all these give 144.

Alternatively, if we have 2 * ( (3 * (4 * 5)) + 1 ) = 122.

Or, if we have (2 * 3) * ( (4 * 5) + 1 ) = 126.

Or, if we have 2 * 3 * ( (4 * 5) + 1 ) = 126.

Or, if we have (2 * 3 * 4 * 5) + 1 = 121.

So, that's all.

Therefore, the different values are 121, 122, 126, 144.

But wait, let me check if there are other ways to group.

For example, if we have (2 * (3 * (4 + 5))) + 1. Wait, but that would require adding 4 and 5, which is not in the original expression.

Wait, the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses to do 4 + 5 is not allowed?

Wait, no. Wait, the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

But inserting parentheses can change the order of operations. For example, if we do 2·(3·(4·(5 + 1))) which is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to do 4 + 5, which is allowed?

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. However, inserting parentheses to do 4 + 5 would require that the addition is between 4 and 5, which is not part of the original expression.

Wait, but the original expression is written as 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses to do 4 + 5 would be changing the original expression.

Wait, but the problem says: "we can obtain values other than 121 for this expression if we are allowed to change it by inserting parentheses."

So, inserting parentheses is allowed, but we cannot rearrange terms.

Therefore, the original expression is 2·3·4·5 + 1. So, the numbers are in the order 2, 3, 4, 5, and then +1.

Therefore, inserting parentheses can only group the existing operations.

Therefore, the addition is only between 5 and 1. Therefore, inserting parentheses cannot introduce a new addition between 4 and 5.

Therefore, the only addition is between 5 and 1.

Therefore, the previous values of 121, 122, 126, 144 are the only possible ones.

But let me check again.

Suppose we have 2·(3·(4·(5 + 1))) = 144.

Or, 2·(3·( (4·5) + 1 )) = 122.

Or, (2·3)·( (4·5) + 1 ) = 126.

Or, (2·3·4·5) + 1 = 121.

Are there any other ways?

For example, if we have (2·(3·4))·(5 + 1) = 144.

Or, ((2·3)·4)·(5 + 1) = 144.

Or, 2·( (3·4)·(5 + 1) ) = 144.

So, all these give 144.

Alternatively, if we have 2·(3·(4·5 + 1)) = 126.

Wait, no, that's 2·(3·(20 + 1)) = 2·(3·21) = 126.

Wait, but that is different from 2·(3·( (4·5) + 1 )) which is same as above.

So, that's 126.

Alternatively, if we have (2·3·(4·5)) + 1 = 120 + 1 = 121.

So, same as original.

Alternatively, if we have 2·( (3·4·5) + 1 ) = 2·(60 + 1) = 122.

So, that's another value.

Therefore, the different values are 121, 122, 126, 144.

So, that's four different values.

But let me check if there are more.

Suppose we have ((2·3)·(4·5)) + 1 = 6·20 + 1 = 121.

Same as original.

Alternatively, (2·(3·4))·(5 + 1) = 24·6 = 144.

Alternatively, (2·(3·(4·5))) + 1 = 120 + 1 = 121.

Alternatively, 2·( (3·(4·5)) + 1 ) = 2·(60 + 1) = 122.

Alternatively, 2·(3·(4·(5 + 1))) = 144.

Alternatively, (2·3)·(4·(5 + 1)) = 6·24 = 144.

Alternatively, 2·(3·(4·(5 + 1))) = 144.

Alternatively, 2·( (3·4)·(5 + 1) ) = 144.

Alternatively, ( (2·3·4)·5 ) + 1 = 24·5 + 1 = 120 + 1 = 121.

So, no new values.

Therefore, the only different values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once more.

Is there a way to get another value?

For example, if we have (2 + 3)·4·5 + 1? But that would require rearranging terms, which is not allowed.

The problem says: "Note that rearranging terms is not allowed, only inserting parentheses."

Therefore, we cannot rearrange the terms.

Therefore, the numbers must remain in the order 2, 3, 4, 5, then +1.

Therefore, the only possible values are 121, 122, 126, 144.

Therefore, the total number of different values is 4.

But wait, let me check if 122 and 126 are distinct.

Yes, 122 is different from 126.

So, total of four different values.

But let me check once again.

Original value: 121.

Other values:

- 122: 2*(3*4*5 + 1) = 2*(60 + 1) = 122.

- 126: 2*(3*(4*5 + 1)) = 2*(3*21) = 126.

- 144: multiple ways, as discussed.

So, yes, four different values.

Therefore, the answer is 4.

But wait, let me check if there's another value.

Suppose we have (2·3·(4 + 5)) + 1. But that would be (2·3·9) + 1 = 54 + 1 = 55.

But is this allowed?

Wait, the original expression is 2·3·4·5 + 1.

If we insert parentheses to do (2·3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed?

But the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1.

But inserting parentheses to do 4 + 5 would require that the addition is between 4 and 5, which is not part of the original expression.

Wait, but the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

Therefore, inserting parentheses to do 4 + 5 would require changing the order of operations, but since the original expression doesn't have an addition between 4 and 5, is that allowed?

Wait, the problem says: "inserting parentheses". So, inserting parentheses can change the order of operations, but the original expression has only one addition.

Therefore, inserting parentheses can only group existing operations.

But the original expression has only one addition. Therefore, inserting parentheses cannot introduce a new addition.

Therefore, the addition is only between 5 and 1. Therefore, inserting parentheses cannot introduce an addition between 4 and 5.

Therefore, the expression (2·3·(4 + 5)) + 1 is not allowed, because it introduces a new addition between 4 and 5.

Therefore, that value of 55 is not allowed.

Therefore, the only allowed values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once more.

Suppose we have 2·(3·(4·(5 + 1))) = 144.

Or, 2·( (3·4)·(5 + 1) ) = 144.

Or, (2·3)·(4·(5 + 1)) = 144.

Or, ((2·3)·4)·(5 + 1) = 144.

Or, (2·(3·4))·(5 + 1) = 144.

So, all these give 144.

Therefore, only four different values.

Therefore, the answer is 4.

But wait, the example given in the problem is (2·(3·4))·(5 + 1) = 144.

So, that's one of the ways.

Therefore, the total number of different values is 4.

But let me check once again if there are other possibilities.

Suppose we have 2·3·(4·(5 + 1)) = 6·24 = 144.

Or, 2·(3·4·(5 + 1)) = 2·(3·4·6) = 2·72 = 144.

Or, (2·3·4)·(5 + 1) = 24·6 = 144.

So, all these give 144.

Therefore, no new values.

Therefore, the answer is 4.

But wait, let me check if there's a way to get 121 again.

Yes, original expression.

So, total of four different values.

Therefore, the answer is 4.

But wait, the problem says "how many values can be obtained from the expression... by inserting parentheses".

But let me check once again if there are other values.

For example, if we have ((2·3)·(4·5)) + 1 = 6·20 + 1 = 121.

Same as original.

Or, (2·(3·4·5)) + 1 = 120 + 1 = 121.

Same.

Or, 2·( (3·4·5) + 1 ) = 122.

Or, 2·(3·(4·5 + 1)) = 126.

Or, 2·(3·(4·(5 + 1))) = 144.

Or, (2·3)·(4·(5 + 1)) = 144.

So, no other values.

Therefore, the answer is 4.

But wait, I remember that sometimes with parentheses, you can get different results.

Wait, let me think of another way.

Suppose we have 2·(3·(4·(5 + 1))) = 144.

Or, 2·( (3·4)·(5 + 1) ) = 144.

Or, (2·3)·( (4·5) + 1 ) = 126.

Or, 2·( (3·(4·5)) + 1 ) = 122.

Or, (2·3·4·5) + 1 = 121.

So, that's five different expressions, but only four different values.

Yes, 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check if there's another value.

Suppose we have (2·(3·(4 + 5))) + 1. But as discussed earlier, this would require adding 4 and 5, which is not allowed.

But if we consider that the original expression is 2·3·4·5 + 1, and we can insert parentheses to do 2·(3·(4 + 5)) + 1, but that would be changing the original expression.

But since the original expression has no addition between 4 and 5, inserting parentheses to do that would be invalid?

Wait, no. Wait, the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

Therefore, inserting parentheses can change the order of operations. For example, if we do 2·(3·(4·(5 + 1))), that is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to do 4 + 5, which is not part of the original expression.

But the original expression has only one addition, between 5 and 1. Therefore, inserting parentheses to do 4 + 5 would require that the addition is between 4 and 5, which is not present.

Therefore, that is not allowed.

Therefore, the value 55 is not allowed.

Therefore, the only values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once again.

Original expression: 2·3·4·5 + 1 = 121.

Other values:

- 2·(3·(4·(5 + 1))) = 144.

- 2·(3·(4·5 + 1)) = 126.

- 2·(3·4·5 + 1) = 122.

- (2·3·4)·(5 + 1) = 144.

- (2·3)·(4·(5 + 1)) = 144.

- 2·( (3·4)·(5 + 1) ) = 144.

- (2·(3·4))·(5 + 1) = 144.

- ( (2·3)·4 )·(5 + 1) = 144.

- 2·( (3·(4·5)) + 1 ) = 122.

- (2·3)·( (4·5) + 1 ) = 126.

- 2·3·( (4·5) + 1 ) = 126.

- (2·3·4·5) + 1 = 121.

So, all these give the same four values.

Therefore, the answer is 4.

But wait, the problem says "how many values can be obtained".

But I need to make sure that there are no other values.

For example, if we have ((2·3)·(4·5)) + 1 = 6·20 + 1 = 121.

Same as original.

Or, ((2·(3·4))·5) + 1 = 24·5 + 1 = 121.

Same.

Or, (2·(3·(4·5))) + 1 = 120 + 1 = 121.

Same.

Therefore, no new values.

Therefore, the answer is 4.

But I need to check if there's a possibility of another value.

Wait, what if we have something like 2·(3 + 4·5) + 1?

But that would be 2·(3 + 20) + 1 = 2·23 + 1 = 46 + 1 = 47.

But again, this requires inserting an addition between 3 and 4·5, which is not part of the original expression.

But the original expression is 2·3·4·5 + 1.

So, inserting parentheses to do 3 + 4·5 would require changing the order of operations, but since the original expression has no addition between 3 and 4, this is not allowed?

Wait, the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

Therefore, inserting parentheses to do 3 + 4·5 would require that the addition is between 3 and 4·5, which is not part of the original expression.

Therefore, this is not allowed.

Therefore, the value 47 is not allowed.

Therefore, the only allowed values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once again.

Is there a way to get 121, 122, 126, 144?

Yes.

Therefore, the total number of different values is 4.

Therefore, the answer is \boxed{4}.

**Final Answer**
\boxed{4}

To determine how many **distinct values** can be obtained from the expression
$$
2 \cdot 3 \cdot 4 \cdot 5 + 1
$$
by **inserting parentheses**, we must consider all valid ways to parenthesize the expression while **preserving the order of the numbers** and **not rearranging them**.

---

### Step 1: Understand the Expression

The original expression is:
$$
2 \cdot 3 \cdot 4 \cdot 5 + 1
$$
According to standard order of operations, multiplication is performed before addition. So the original value is:
$$
(2 \cdot 3 \cdot 4 \cdot 5) + 1 = 120 + 1 = 121
$$

However, by inserting parentheses, we can change the **order of operations**, potentially altering the result.

---

### Step 2: Consider All Valid Parentheses Insertions

We are only allowed to **insert parentheses**, not rearrange the numbers. The expression has:

- Four multiplications: $2 \cdot 3$, $3 \cdot 4$, $4 \cdot 5$
- One addition: $5 + 1$

We can group the multiplications in different ways and apply the addition at different points. The key is to **group the addition with some of the multiplications**.

Let’s explore the different groupings and compute the resulting values:

---

### Step 3: Compute All Possible Values

#### 1. **Original Expression**
$$
(2 \cdot 3 \cdot 4 \cdot 5) + 1 = 120 + 1 = \boxed{121}
$$

#### 2. **Group the addition with 5 and 1, then multiply by 4**
$$
(2 \cdot 3 \cdot 4) \cdot (5 + 1) = 24 \cdot 6 = \boxed{144}
$$

#### 3. **Group the addition with 5 and 1, then multiply by 3**
$$
(2 \cdot 3) \cdot (4 \cdot (5 + 1)) = 6 \cdot 24 = \boxed{144}
$$

#### 4. **Group the addition with 5 and 1, then multiply by 2**
$$
2 \cdot (3 \cdot (4 \cdot (5 + 1))) = 2 \cdot (3 \cdot 24) = 2 \cdot 72 = \boxed{144}
$$

#### 5. **Group the addition with 4 and 5, then multiply by 3 and 2**
$$
2 \cdot (3 \cdot (4 \cdot 5 + 1)) = 2 \cdot (3 \cdot 21) = 2 \cdot 63 = \boxed{126}
$$

#### 6. **Group the addition with 4 and 5, then multiply by 2 and 3**
$$
(2 \cdot 3) \cdot (4 \cdot 5 + 1) = 6 \cdot 21 = \boxed{126}
$$

#### 7. **Group the addition with 3, 4, and 5, then multiply by 2**
$$
2 \cdot (3 \cdot 4 \cdot 5 + 1) = 2 \cdot (60 + 1) = 2 \cdot 61 = \boxed{122}
$$

---

### Step 4: Identify Unique Values

From the above, the **distinct values** obtained are:

- $121$
- $122$
- $126$
- $144$

All other groupings result in one of these values.

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many different values can be obtained from the expression 2·3·4·5 + 1 by inserting parentheses. The original value is 121 because multiplication is done before addition. But if we can change the order of operations by adding parentheses, we might get different results. The example given is (2·(3·4))·(5+1) = 144. So, my task is to find all possible values by inserting parentheses in different ways and count them.

First, let me recall that inserting parentheses can change the order in which operations are performed. Since there are both multiplications and an addition, I need to consider how parentheses can affect the order of these operations.

The original expression is 2·3·4·5 + 1. Let me note that the operations are all multiplications except for the addition at the end. So, the expression is a sequence of multiplications followed by an addition. However, inserting parentheses can group some of the multiplications together or even group the addition with some multiplications.

But wait, the addition is only between the product of 2·3·4·5 and 1. However, if we insert parentheses, we can change when the addition happens. For example, if we group the addition with some of the multiplications, we might get different results.

Let me think about the structure of the expression. The original expression is:

( ( (2·3)·4 )·5 ) + 1 = 121

But if we insert parentheses differently, we can change the order. However, since all the operations are multiplications except for the addition, the addition is only between the entire product of 2·3·4·5 and 1. But if we can insert parentheses to make the addition happen earlier, that would change the value.

Wait, but the original expression is written as 2·3·4·5 + 1. So, according to order of operations, multiplication is done before addition. However, if we insert parentheses, we can make the addition happen before some multiplications. For example, if we do 2·(3·4·5 + 1), that would be different.

So, the key is that the addition is between the product of 2·3·4·5 and 1, but if we can parenthesize to make the addition occur at different points.

But how many different ways can we insert parentheses here? Let me think about the expression as a sequence of operations. Let me consider the expression as a sequence of numbers and operations: 2, ·, 3, ·, 4, ·, 5, +, 1.

So, there are four multiplications and one addition. The question is about inserting parentheses to change the order of operations. However, since multiplication is associative and commutative, but addition is not. However, since we are only allowed to insert parentheses, not rearrange terms, the positions of the numbers are fixed.

So, the problem reduces to determining all possible values that can be obtained by parenthesizing the expression 2·3·4·5 + 1.

Let me think about the different ways to parenthesize. Since there are multiple multiplications and one addition, the addition can be grouped with some of the multiplications.

First, let me note that the addition is between the product of 2·3·4·5 and 1. However, if we can insert parentheses to make the addition occur before some multiplications. For example, if we do 2·(3·(4·(5 + 1))), which would be 2·(3·(4·6)) = 2·(3·24) = 2·72 = 144. Wait, that's the same as the example given.

Alternatively, if we do (2·3)·(4·(5 + 1)) = 6·(4·6) = 6·24 = 144. Same result.

Alternatively, if we do ((2·3)·4)·(5 + 1) = 24·6 = 144. Also same.

So, different parenthesizations can lead to the same value. So, the key is to find all unique values that can be obtained through different parenthesizations.

So, perhaps I need to consider all possible ways to parenthesize the expression, compute the value for each, and then count the unique ones.

But how do I systematically approach this?

First, let me note that the expression has five numbers and four operations: 2, ·, 3, ·, 4, ·, 5, +, 1.

But the addition is only between the product of the first four numbers and 1. However, inserting parentheses can change when the addition is performed.

Alternatively, maybe I can think of the expression as two parts: the product of 2, 3, 4, 5 and the addition with 1. But if we can insert parentheses to make the addition happen earlier.

But since the addition is only between the product and 1, but if we can group some of the multiplications with the addition.

Wait, for example, if we do 2·(3·(4·(5 + 1))) as above. Or (2·(3·(4·5))) + 1, which is the original expression.

Alternatively, maybe grouping the addition with some of the numbers. For example, (2·3·4·(5 + 1)) which is 2·3·4·6 = 144.

Alternatively, ((2·3)·(4·(5 + 1))) = 6·(4·6) = 6·24 = 144.

Alternatively, (2·(3·4·(5 + 1))) = 2·(3·4·6) = 2·72 = 144.

Alternatively, ((2·(3·4))·(5 + 1)) = (2·12)·6 = 24·6 = 144.

So, all these different parenthesizations lead to 144.

But what if we do something else? For example, if we do 2·(3·(4·5 + 1))? Let me compute that.

First, compute 4·5 + 1 = 20 + 1 = 21. Then 3·21 = 63. Then 2·63 = 126.

So that's a different value.

Similarly, if we do (2·3)·(4·5 + 1) = 6·(20 + 1) = 6·21 = 126.

Same result.

Alternatively, 2·(3·4·5 + 1) = 2·(60 + 1) = 2·61 = 122.

Wait, that's different.

So, that's another value.

So, here we have different values: 121, 122, 126, 144.

Wait, but maybe there are more. Let me check.

Let me think of all possible ways to insert parentheses.

First, the original expression is ((2·3)·4)·5 + 1 = 121.

But if we can insert parentheses to make the addition happen at different points.

So, the addition is between the product of the first four numbers and 1. But if we can insert parentheses to make the addition happen earlier.

So, the addition can be grouped with some of the multiplications.

Let me think of the expression as a combination of multiplications and an addition.

Let me consider the addition as an operation that can be applied to different parts.

So, the expression is: [expression] + 1, where [expression] is 2·3·4·5. But if we can parenthesize to have the addition occur earlier.

Alternatively, maybe the addition is between some of the multiplications and 1.

So, for example, if we have 2·(3·(4·(5 + 1))) as before.

Or, (2·(3·4))·(5 + 1).

Or, 2·( (3·4)·(5 + 1) ).

Or, (2·3)·(4·(5 + 1)).

Or, ((2·3)·4)·(5 + 1).

Or, 2·(3·(4·5 + 1)).

Or, (2·(3·(4·5))) + 1.

Wait, but that's the original expression.

So, perhaps there are different ways to parenthesize the addition.

Let me think of the addition as an operation that can be inserted between different parts.

So, the original expression is ( ( (2·3)·4 )·5 ) + 1.

But if we can insert parentheses to make the addition happen between different terms.

So, for example, if we do 2·3·(4·(5 + 1)) which is 2·3·(4·6) = 2·3·24 = 144.

Or, 2·(3·4·(5 + 1)) = 2·(3·4·6) = 2·72 = 144.

Or, (2·3·4)·(5 + 1) = 24·6 = 144.

Or, 2·( (3·4)·(5 + 1) ) = same as above.

Alternatively, if we do (2·(3·(4·5))) + 1 = 2·(3·20) + 1 = 2·60 + 1 = 121.

Alternatively, if we do 2·(3·(4·5 + 1)) = 2·(3·(20 + 1)) = 2·(3·21) = 2·63 = 126.

Alternatively, (2·3)·(4·(5 + 1)) = 6·(4·6) = 6·24 = 144.

Alternatively, (2·(3·4))·(5 + 1) = (2·12)·6 = 24·6 = 144.

Alternatively, ((2·3)·4)·(5 + 1) = 24·6 = 144.

Alternatively, 2·(3·(4·(5 + 1))) = 2·(3·(4·6)) = 2·(3·24) = 2·72 = 144.

Alternatively, 2·( (3·4)·(5 + 1) ) = same as above.

Alternatively, ( (2·(3·4))·5 ) + 1 = (2·12·5) + 1 = 120 + 1 = 121.

Wait, that's the original expression.

Alternatively, if we do (2·(3·(4·5))) + 1 = same as original.

Alternatively, if we do ( (2·3)·(4·5) ) + 1 = same as original.

Alternatively, if we do 2·(3·4·5 + 1) = 2·(60 + 1) = 2·61 = 122.

So, that's another value.

So, so far, I have found values: 121, 122, 126, 144.

Are there more? Let me check.

What if we do something like (2·3·(4·5)) + 1 = same as original.

Alternatively, 2·(3·(4·5 + 1)) = 126.

Alternatively, 2·(3·4·(5 + 1)) = 144.

Alternatively, 2·( (3·4)·(5 + 1) ) = same.

Alternatively, (2·(3·4))·(5 + 1) = 144.

Alternatively, (2·(3·(4·5))) + 1 = 121.

Alternatively, ( (2·3)·(4·(5 + 1)) ) = 144.

Alternatively, ( (2·(3·4))·(5 + 1) ) = 144.

Alternatively, ( ( (2·3)·4 )·(5 + 1) ) = 144.

Alternatively, 2·( (3·(4·5)) + 1 ) = 2·(60 + 1) = 122.

Wait, that's the same as before.

Alternatively, if we do (2·3)·(4·5 + 1) = 6·(20 + 1) = 6·21 = 126.

So, same as before.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do 2·( (3·4)·(5 + 1) ) = 144.

Is there a way to get another value? Let me think.

What if we do something like (2·(3·(4 + 5)))·1? Wait, but the original expression is 2·3·4·5 + 1. So, the numbers are fixed. The addition is only between the product and 1. Wait, but if we can insert parentheses to make the addition happen between different numbers.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product of 2·3·4·5 and 1. However, if we can insert parentheses to make the addition happen between some of the multiplications and 1.

Wait, for example, if we do 2·(3·4·5 + 1). That is, adding 1 to the product of 3·4·5 first, then multiplying by 2.

Similarly, if we do (2·3·4)·(5 + 1) = 24·6 = 144.

Alternatively, if we do (2·3)·(4·(5 + 1)) = 6·24 = 144.

Alternatively, if we do 2·( (3·4)·(5 + 1) ) = same.

Alternatively, if we do 2·(3·(4·(5 + 1))) = same.

Alternatively, if we do (2·(3·(4·5))) + 1 = 121.

Alternatively, if we do ( (2·3)·4 )·(5 + 1) = 24·6 = 144.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do (2·(3·4·5)) + 1 = 121.

Alternatively, if we do 2·(3·4·(5 + 1)) = 144.

Alternatively, if we do (2·3)·(4·5 + 1) = 6·21 = 126.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do (2·(3·(4·5))) + 1 = 121.

Alternatively, if we do (2·(3·4))·(5 + 1) = 24·6 = 144.

So, so far, the values I have are 121, 122, 126, 144.

Wait, where does 122 come from? That was when we did 2·(3·4·5 + 1) = 2·(60 + 1) = 122.

Is there another way to get a different value? Let me think.

What if we do something like (2·3·(4·5)) + 1 = same as original.

Alternatively, if we do 2·(3·(4·5 + 1)) = 126.

Alternatively, if we do (2·(3·(4 + 5)))·1? Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. If we try to add 4 and 5 first, that would be changing the original expression. Wait, but the original expression is 2·3·4·5 + 1. So, the numbers are fixed. So, inserting parentheses can't change the order of the numbers.

Wait, but if we do 2·(3·(4 + 5))·1? Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. So, inserting parentheses can't change the positions of the numbers.

Wait, but if we do 2·(3·(4 + 5)) + 1? Wait, that would be different. Let me check.

But the original expression is 2·3·4·5 + 1. If we insert parentheses to make it 2·(3·(4 + 5)) + 1, that would be allowed? Wait, but the original expression is written as 2·3·4·5 + 1. So, inserting parentheses can only group operations, not change the order of numbers.

Wait, but if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. But in the original expression, the addition is only between the product and 1. However, inserting parentheses can allow the addition to happen between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product of 2·3·4·5 and 1. However, inserting parentheses can change the order of operations. For example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, if we do 2·(3·(4·(5 + 1))), that is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. However, in the original expression, there is no addition between 4 and 5. Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is written as 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition. So, the addition is only between the product of the first four numbers and 1. However, inserting parentheses can change the order of operations. For example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, if we do 2·(3·(4·(5 + 1))), that is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. However, in the original expression, there is no addition between 4 and 5. Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product of 2·3·4·5 and 1. However, if we insert parentheses to make the addition happen between 4 and 5, that would be changing the original expression. Wait, but the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Let me think. If we have 2·3·4·5 + 1, inserting parentheses can change the order. For example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, if we do 2·(3·(4·(5 + 1))). But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed. However, in the original expression, there is no addition between 4 and 5. Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the entire product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Because the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product of 2·3·4·5 and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Because the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, but how? Because the original expression is written as 2·3·4·5 + 1. So, the addition is only between the product and 1. However, inserting parentheses can allow us to perform the addition between 4 and 5.

Wait, maybe I need to think of the expression as having operations between numbers. Let me write down the expression with all operations:

2 * 3 * 4 * 5 + 1

So, the operations are:

Between 2 and 3: multiplication

Between 3 and 4: multiplication

Between 4 and 5: multiplication

Between 5 and 1: addition

So, the addition is only between 5 and 1. Therefore, inserting parentheses can only group the multiplications and the addition.

Therefore, the addition is only between 5 and 1. Therefore, inserting parentheses can't change the fact that the addition is between 5 and 1. Wait, but that seems contradictory to the example given.

Wait, the example given is (2·(3·4))·(5 + 1) = 144. So, in this case, the addition is between 5 and 1, but they are grouped with the multiplication.

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is between the product of 2·3·4·5 and 1. However, inserting parentheses can group the addition with some of the multiplications.

Wait, but the addition is only between 5 and 1. So, if we want to group the addition with some multiplications, we need to have parentheses around 5 + 1, and then multiply that with other terms.

So, for example, if we do 2·3·(4·(5 + 1)), that is allowed. Similarly, (2·3·4)·(5 + 1).

But if we want to do something like 2·(3·(4 + 5)) + 1, that would require changing the order of operations, but in the original expression, the addition is only between 5 and 1. Therefore, inserting parentheses can't change the fact that the addition is between 5 and 1. Wait, but how is the example given possible?

Wait, the example is (2·(3·4))·(5 + 1). So, that is grouping the multiplication of 3 and 4, then multiplying by 2, then multiplying by (5 + 1). So, the addition is still between 5 and 1.

Therefore, the addition is always between 5 and 1. Therefore, inserting parentheses can only group the multiplications and the addition.

Therefore, the addition is fixed between 5 and 1. Therefore, the only way to change the value is by grouping the multiplications in different ways, and then combining with the addition.

Wait, but then how can we get 126? For example, 2·(3·(4·5 + 1)) = 2·(3·(20 + 1)) = 2·(3·21) = 126.

But in this case, the addition is between 5 and 1, but we have 4·5 + 1. Wait, but that would require that the addition is between 5 and 1, but also between 4 and 5?

Wait, no. Wait, if we have 4·5 + 1, that would be 4·5 + 1, which is 20 + 1 = 21. But in the original expression, the addition is only between 5 and 1. Therefore, inserting parentheses to do 4·(5 + 1) is allowed, because that is grouping the addition between 5 and 1 with the multiplication by 4.

Wait, so the addition is between 5 and 1, but we can group that addition with previous multiplications.

Therefore, the addition is fixed between 5 and 1, but we can group that addition with previous multiplications.

Therefore, the different values come from different ways of grouping the multiplications before the addition.

So, the expression can be thought of as (some product of 2, 3, 4, 5) + 1, where the product can be grouped in different ways.

Wait, but if we do 2·(3·(4·(5 + 1))), that is grouping the addition with 5 and 1, then multiplying by 4, then by 3, then by 2.

Alternatively, if we do (2·3·4)·(5 + 1), that is grouping the addition with 5 and 1, then multiplying by 2·3·4.

Alternatively, if we do 2·(3·(4·5 + 1)), that is grouping the addition with 5 and 1, but also multiplying by 4? Wait, no. Wait, 4·5 + 1 is 21, then multiplied by 3, then multiplied by 2.

Wait, but in this case, the addition is between 5 and 1, but we have 4·(5 + 1) or (4·5) + 1?

Wait, if we have 4·(5 + 1), that is grouping the addition with 5 and 1, then multiplying by 4.

If we have (4·5) + 1, that is multiplying 4 and 5 first, then adding 1.

So, depending on how we group the operations, we can have different results.

Therefore, the key is that the addition is between 5 and 1, but we can choose to group that addition with previous multiplications or not.

Therefore, the different values come from different ways of grouping the multiplications before the addition.

So, the problem reduces to: how many different values can be obtained by grouping the multiplications of 2, 3, 4, 5 in different ways, and then adding 1.

Wait, but the addition is only between the product and 1. However, if we group the addition with some multiplications, we can have different results.

Wait, but if we think of the entire expression as (some expression involving 2, 3, 4, 5) + 1, where the "some expression" can be grouped in different ways.

Therefore, the number of different values depends on the number of different ways to parenthesize the multiplications of 2, 3, 4, 5.

But since multiplication is associative, different parenthesizations of the same product give the same result. Wait, but if we have different groupings, but since multiplication is associative, they all give the same result.

Wait, but in our previous examples, we saw that different parenthesizations can lead to different results when combined with the addition.

Wait, for example, if we have (2·3·4·5) + 1 = 120 + 1 = 121.

But if we have 2·(3·(4·(5 + 1))) = 2·(3·(4·6)) = 2·(3·24) = 144.

But here, the addition is between 5 and 1, but we have grouped the multiplication with 4 and the addition.

So, in this case, the product is not the entire product of 2, 3, 4, 5, but rather, some of the multiplications are combined with the addition.

Therefore, the key is that the addition can be combined with some of the multiplications.

Therefore, the problem is not just about parenthesizing the multiplications, but also about when the addition is applied.

Therefore, the addition can be applied at different points.

So, the addition is between 5 and 1, but we can choose to apply it before multiplying by 4, or before multiplying by 3, etc.

Therefore, the different values come from different ways of inserting the addition into the multiplication sequence.

So, for example, if we consider the expression as a sequence of operations: 2 * 3 * 4 * 5 + 1.

We can think of the addition as an operation that can be inserted at different positions.

But since the addition is only between 5 and 1, we can think of inserting parentheses to group the addition with some of the multiplications.

So, the addition can be grouped with the last multiplication (i.e., 5 + 1), or with the previous multiplication (i.e., 4*(5 + 1)), or with the previous previous multiplication (i.e., 3*(4*(5 + 1))), etc.

Therefore, the number of different values depends on how many different ways we can insert the addition into the multiplication sequence.

So, for example, the addition can be inserted after the first multiplication (i.e., 2*(3*(4*(5 + 1)))), after the second multiplication (i.e., (2*3)*(4*(5 + 1))), after the third multiplication (i.e., ((2*3)*4)*(5 + 1)), or after the fourth multiplication (i.e., (((2*3)*4)*5) + 1).

Wait, but inserting the addition after the fourth multiplication is the original expression.

But inserting the addition after the third multiplication would be ((2*3)*4)*(5 + 1) = 24*6 = 144.

Inserting the addition after the second multiplication would be (2*3)*(4*(5 + 1)) = 6*24 = 144.

Inserting the addition after the first multiplication would be 2*(3*(4*(5 + 1))) = 2*(3*24) = 144.

So, all these give the same value.

But earlier, we saw that if we do 2*(3*(4*5 + 1)) = 2*(3*21) = 126.

Wait, how is that different?

Ah, here, the addition is between 5 and 1, but we have 4*5 + 1 instead of 4*(5 + 1).

So, this is different.

Therefore, the addition can be applied at different points.

So, the addition can be applied either before or after some multiplications.

Wait, but how?

Wait, the original expression is 2·3·4·5 + 1.

If we want to apply the addition before multiplying by 4, we need to have 4*(5 + 1).

If we want to apply the addition before multiplying by 3, we need to have 3*(4*(5 + 1)).

If we want to apply the addition before multiplying by 2, we need to have 2*(3*(4*(5 + 1))).

Alternatively, if we apply the addition after multiplying by 4, we have (4*5) + 1.

Similarly, applying the addition after multiplying by 3: (3*4*5) + 1.

Wait, but how does that work?

Wait, if we have 2*(3*(4*5) + 1).

So, that would be 2*( (3*(4*5)) + 1 ) = 2*(60 + 1) = 122.

Similarly, if we have (2*3)*(4*5 + 1) = 6*(20 + 1) = 6*21 = 126.

So, here, the addition is applied after multiplying by 4, but before multiplying by 3 or 2.

Therefore, the addition can be applied at different points in the multiplication sequence.

Therefore, the different values come from different positions where the addition is applied.

Therefore, the problem reduces to determining how many different positions we can apply the addition, and what values that gives.

So, the addition is between 5 and 1.

But we can choose to apply the addition at different points.

Let me think of the expression as follows:

We have the numbers 2, 3, 4, 5, and then +1.

The operations between the numbers are all multiplications, except for the last operation which is addition.

But inserting parentheses can change when the addition is applied.

So, the addition can be applied after multiplying 2, 3, 4, or 5.

Wait, but how?

Let me think of the expression as a sequence of operations.

Let me consider the expression as follows:

Start with 2, then multiply by 3, then multiply by 4, then multiply by 5, then add 1.

But inserting parentheses can change the order.

For example, if we add parentheses to apply the addition before multiplying by 4, we get:

2 * 3 * (4 * (5 + 1))

Which is 2 * 3 * (4 * 6) = 2 * 3 * 24 = 144.

Alternatively, if we apply the addition before multiplying by 3:

2 * (3 * (4 * (5 + 1))) = same as above.

Alternatively, if we apply the addition before multiplying by 2:

2 * (3 * (4 * (5 + 1))) = same.

Alternatively, if we apply the addition after multiplying by 5, which is the original expression:

(2 * 3 * 4 * 5) + 1 = 121.

Alternatively, if we apply the addition after multiplying by 4:

(2 * 3 * 4) * (5 + 1) = 24 * 6 = 144.

Alternatively, if we apply the addition after multiplying by 3:

(2 * 3) * (4 * (5 + 1)) = 6 * 24 = 144.

Alternatively, if we apply the addition after multiplying by 2:

2 * (3 * (4 * (5 + 1))) = 144.

Alternatively, if we apply the addition between multiplying by 4 and 5, but that would be (4 * 5) + 1, which is 21. Then, multiplying by 3 and 2.

So, for example:

2 * (3 * ( (4 * 5) + 1 )) = 2 * (3 * 21) = 126.

Similarly, (2 * 3) * ( (4 * 5) + 1 ) = 6 * 21 = 126.

Alternatively, 2 * 3 * ( (4 * 5) + 1 ) = 6 * 21 = 126.

Alternatively, if we apply the addition after multiplying by 5, but that's the original expression.

Alternatively, if we apply the addition after multiplying by 3 and 4, but that would be ( (2 * 3) * 4 ) * (5 + 1) = 24 * 6 = 144.

Alternatively, if we apply the addition after multiplying by 2, 3, and 4, which is the original expression.

Alternatively, if we apply the addition after multiplying by 2 and 3, which is ( (2 * 3) * 4 * 5 ) + 1 = 120 + 1 = 121.

Alternatively, if we apply the addition after multiplying by 2, which would be 2 * ( (3 * 4 * 5) + 1 ) = 2 * (60 + 1) = 122.

So, here, applying the addition after multiplying by 2, but before multiplying by 3, 4, 5.

Wait, but how is that possible?

If we have 2 * ( (3 * 4 * 5) + 1 ), that would mean that we first compute 3 * 4 * 5, then add 1, then multiply by 2.

But in the original expression, the addition is between 5 and 1. So, inserting parentheses to do 3 * 4 * 5 + 1 is allowed?

Wait, but the original expression is 2 * 3 * 4 * 5 + 1. So, the addition is only between 5 and 1. However, inserting parentheses to do 3 * 4 * 5 + 1 is allowed, because that is grouping the addition with the previous multiplications.

Therefore, the addition can be applied at different points in the multiplication sequence.

So, the different values come from different positions where the addition is applied.

So, the addition can be applied after multiplying by 2, 3, 4, or 5.

Wait, but how?

Let me think of the expression as follows:

The addition is between 5 and 1.

But we can choose to apply the addition at different points by grouping.

For example:

Case 1: Apply the addition after multiplying by 5: (2 * 3 * 4 * 5) + 1 = 121.

Case 2: Apply the addition after multiplying by 4: (2 * 3 * 4) * (5 + 1) = 24 * 6 = 144.

Case 3: Apply the addition after multiplying by 3: (2 * 3) * (4 * (5 + 1)) = 6 * 24 = 144.

Case 4: Apply the addition after multiplying by 2: 2 * (3 * (4 * (5 + 1))) = 2 * (3 * 24) = 144.

Case 5: Apply the addition between multiplying by 4 and 5: 2 * 3 * ( (4 * 5) + 1 ) = 6 * 21 = 126.

Case 6: Apply the addition between multiplying by 3 and 4: 2 * (3 * (4 * 5) + 1 ) = 2 * (60 + 1) = 122.

Case 7: Apply the addition between multiplying by 2 and 3: (2 * (3 * 4 * 5)) + 1 = 120 + 1 = 121.

Wait, but case 7 is same as case 1.

Similarly, case 5 and case 6 are different.

So, so far, we have values: 121, 122, 126, 144.

Are there more cases?

Wait, can we apply the addition in more positions?

For example, if we apply the addition between multiplying by 2 and 3, but also between multiplying by 4 and 5?

Wait, but the addition is only between 5 and 1.

Wait, but if we have something like 2 * ( (3 * 4) * (5 + 1) ), which is case 3.

Alternatively, if we have (2 * (3 * (4 * (5 + 1)))) which is case 4.

Alternatively, if we have ( (2 * 3) * (4 * (5 + 1)) ) which is case 3.

Alternatively, if we have (2 * 3) * (4 * (5 + 1)) = same as case 3.

Alternatively, if we have 2 * (3 * (4 * (5 + 1))) = case 4.

Alternatively, if we have 2 * ( (3 * 4) * (5 + 1) ) = same as case 3.

So, all these give 144.

Alternatively, if we have 2 * (3 * ( (4 * 5) + 1 )) = case 6: 122.

Alternatively, if we have (2 * 3) * ( (4 * 5) + 1 ) = case 5: 126.

Alternatively, if we have 2 * 3 * ( (4 * 5) + 1 ) = same as case 5: 126.

Alternatively, if we have (2 * (3 * 4 * 5)) + 1 = case 1: 121.

Alternatively, if we have ( (2 * 3 * 4) * 5 ) + 1 = case 1: 121.

Alternatively, if we have ( (2 * 3) * 4 * 5 ) + 1 = case 1: 121.

So, seems like the only different values are 121, 122, 126, 144.

But let me check if there are other possibilities.

For example, can we apply the addition in more than one place?

Wait, the addition is only between 5 and 1. So, we can't have multiple additions.

But if we have something like 2 * (3 + (4 * 5)) + 1? Wait, but that would require changing the original expression.

Wait, the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses can't introduce new additions.

Therefore, the only addition is between 5 and 1.

Therefore, the only way to apply the addition is either before multiplying by 4, 3, 2, or after multiplying by 5.

Wait, but how?

Wait, if we apply the addition after multiplying by 5, that's the original expression.

If we apply the addition before multiplying by 4, that is grouping 5 + 1 with 4.

If we apply the addition before multiplying by 3, that is grouping 5 + 1 with 4 and 3.

If we apply the addition before multiplying by 2, that is grouping 5 + 1 with 4, 3, and 2.

Alternatively, if we apply the addition between multiplying by 4 and 5, that is grouping 4 * 5 + 1.

Similarly, applying the addition between multiplying by 3 and 4, that is grouping 3 * (4 * 5) + 1.

Wait, but how is that different?

Wait, if we have 2 * (3 * (4 * 5) + 1) = 2 * (60 + 1) = 122.

Alternatively, if we have 2 * 3 * (4 * 5 + 1) = 6 * 21 = 126.

Alternatively, if we have (2 * 3) * (4 * 5 + 1) = 6 * 21 = 126.

Alternatively, if we have 2 * (3 * 4 * 5 + 1) = 2 * 61 = 122.

So, these are two different values: 122 and 126.

So, in total, we have:

- Applying the addition after multiplying by 5: 121.

- Applying the addition after multiplying by 4: 144.

- Applying the addition after multiplying by 3: 144.

- Applying the addition after multiplying by 2: 144.

- Applying the addition between multiplying by 4 and 5: 126.

- Applying the addition between multiplying by 3 and 4: 122.

Wait, but is there another way to apply the addition?

For example, applying the addition between multiplying by 2 and 3?

But that would be (2 * (3 * 4 * 5)) + 1 = 120 + 1 = 121, which is same as case 1.

Alternatively, applying the addition between multiplying by 2 and 3, but also between multiplying by 4 and 5?

But that would require multiple additions, which isn't allowed.

Therefore, the only different values are 121, 122, 126, 144.

Wait, but let me check if there are other possibilities.

For example, if we have something like (2 * (3 + 4)) * (5 + 1). Wait, but that would require changing the original expression.

Wait, the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses can't introduce new additions.

Therefore, the only addition is between 5 and 1.

Therefore, the only different values are 121, 122, 126, 144.

But let me check if there are more possibilities.

For example, if we have 2 * ( (3 * 4) * (5 + 1) ) = 144.

Or, if we have ( (2 * 3) * 4 ) * (5 + 1) = 144.

Or, if we have 2 * (3 * (4 * (5 + 1))) = 144.

So, all these give 144.

Alternatively, if we have 2 * ( (3 * (4 * 5)) + 1 ) = 122.

Or, if we have (2 * 3) * ( (4 * 5) + 1 ) = 126.

Or, if we have 2 * 3 * ( (4 * 5) + 1 ) = 126.

Or, if we have (2 * 3 * 4 * 5) + 1 = 121.

So, that's all.

Therefore, the different values are 121, 122, 126, 144.

But wait, let me check if there are other ways to group.

For example, if we have (2 * (3 * (4 + 5))) + 1. Wait, but that would require adding 4 and 5, which is not in the original expression.

Wait, the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses to do 4 + 5 is not allowed?

Wait, no. Wait, the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

But inserting parentheses can change the order of operations. For example, if we do 2·(3·(4·(5 + 1))) which is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to do 4 + 5, which is allowed?

Wait, but the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1. However, inserting parentheses to do 4 + 5 would require that the addition is between 4 and 5, which is not part of the original expression.

Wait, but the original expression is written as 2·3·4·5 + 1. So, the addition is only between 5 and 1. Therefore, inserting parentheses to do 4 + 5 would be changing the original expression.

Wait, but the problem says: "we can obtain values other than 121 for this expression if we are allowed to change it by inserting parentheses."

So, inserting parentheses is allowed, but we cannot rearrange terms.

Therefore, the original expression is 2·3·4·5 + 1. So, the numbers are in the order 2, 3, 4, 5, and then +1.

Therefore, inserting parentheses can only group the existing operations.

Therefore, the addition is only between 5 and 1. Therefore, inserting parentheses cannot introduce a new addition between 4 and 5.

Therefore, the only addition is between 5 and 1.

Therefore, the previous values of 121, 122, 126, 144 are the only possible ones.

But let me check again.

Suppose we have 2·(3·(4·(5 + 1))) = 144.

Or, 2·(3·( (4·5) + 1 )) = 122.

Or, (2·3)·( (4·5) + 1 ) = 126.

Or, (2·3·4·5) + 1 = 121.

Are there any other ways?

For example, if we have (2·(3·4))·(5 + 1) = 144.

Or, ((2·3)·4)·(5 + 1) = 144.

Or, 2·( (3·4)·(5 + 1) ) = 144.

So, all these give 144.

Alternatively, if we have 2·(3·(4·5 + 1)) = 126.

Wait, no, that's 2·(3·(20 + 1)) = 2·(3·21) = 126.

Wait, but that is different from 2·(3·( (4·5) + 1 )) which is same as above.

So, that's 126.

Alternatively, if we have (2·3·(4·5)) + 1 = 120 + 1 = 121.

So, same as original.

Alternatively, if we have 2·( (3·4·5) + 1 ) = 2·(60 + 1) = 122.

So, that's another value.

Therefore, the different values are 121, 122, 126, 144.

So, that's four different values.

But let me check if there are more.

Suppose we have ((2·3)·(4·5)) + 1 = 6·20 + 1 = 121.

Same as original.

Alternatively, (2·(3·4))·(5 + 1) = 24·6 = 144.

Alternatively, (2·(3·(4·5))) + 1 = 120 + 1 = 121.

Alternatively, 2·( (3·(4·5)) + 1 ) = 2·(60 + 1) = 122.

Alternatively, 2·(3·(4·(5 + 1))) = 144.

Alternatively, (2·3)·(4·(5 + 1)) = 6·24 = 144.

Alternatively, 2·(3·(4·(5 + 1))) = 144.

Alternatively, 2·( (3·4)·(5 + 1) ) = 144.

Alternatively, ( (2·3·4)·5 ) + 1 = 24·5 + 1 = 120 + 1 = 121.

So, no new values.

Therefore, the only different values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once more.

Is there a way to get another value?

For example, if we have (2 + 3)·4·5 + 1? But that would require rearranging terms, which is not allowed.

The problem says: "Note that rearranging terms is not allowed, only inserting parentheses."

Therefore, we cannot rearrange the terms.

Therefore, the numbers must remain in the order 2, 3, 4, 5, then +1.

Therefore, the only possible values are 121, 122, 126, 144.

Therefore, the total number of different values is 4.

But wait, let me check if 122 and 126 are distinct.

Yes, 122 is different from 126.

So, total of four different values.

But let me check once again.

Original value: 121.

Other values:

- 122: 2*(3*4*5 + 1) = 2*(60 + 1) = 122.

- 126: 2*(3*(4*5 + 1)) = 2*(3*21) = 126.

- 144: multiple ways, as discussed.

So, yes, four different values.

Therefore, the answer is 4.

But wait, let me check if there's another value.

Suppose we have (2·3·(4 + 5)) + 1. But that would be (2·3·9) + 1 = 54 + 1 = 55.

But is this allowed?

Wait, the original expression is 2·3·4·5 + 1.

If we insert parentheses to do (2·3·(4 + 5)) + 1, that would be inserting parentheses to group 4 + 5, which is allowed?

But the original expression is 2·3·4·5 + 1. So, the addition is only between 5 and 1.

But inserting parentheses to do 4 + 5 would require that the addition is between 4 and 5, which is not part of the original expression.

Wait, but the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

Therefore, inserting parentheses to do 4 + 5 would require changing the order of operations, but since the original expression doesn't have an addition between 4 and 5, is that allowed?

Wait, the problem says: "inserting parentheses". So, inserting parentheses can change the order of operations, but the original expression has only one addition.

Therefore, inserting parentheses can only group existing operations.

But the original expression has only one addition. Therefore, inserting parentheses cannot introduce a new addition.

Therefore, the addition is only between 5 and 1. Therefore, inserting parentheses cannot introduce an addition between 4 and 5.

Therefore, the expression (2·3·(4 + 5)) + 1 is not allowed, because it introduces a new addition between 4 and 5.

Therefore, that value of 55 is not allowed.

Therefore, the only allowed values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once more.

Suppose we have 2·(3·(4·(5 + 1))) = 144.

Or, 2·( (3·4)·(5 + 1) ) = 144.

Or, (2·3)·(4·(5 + 1)) = 144.

Or, ((2·3)·4)·(5 + 1) = 144.

Or, (2·(3·4))·(5 + 1) = 144.

So, all these give 144.

Therefore, only four different values.

Therefore, the answer is 4.

But wait, the example given in the problem is (2·(3·4))·(5 + 1) = 144.

So, that's one of the ways.

Therefore, the total number of different values is 4.

But let me check once again if there are other possibilities.

Suppose we have 2·3·(4·(5 + 1)) = 6·24 = 144.

Or, 2·(3·4·(5 + 1)) = 2·(3·4·6) = 2·72 = 144.

Or, (2·3·4)·(5 + 1) = 24·6 = 144.

So, all these give 144.

Therefore, no new values.

Therefore, the answer is 4.

But wait, let me check if there's a way to get 121 again.

Yes, original expression.

So, total of four different values.

Therefore, the answer is 4.

But wait, the problem says "how many values can be obtained from the expression... by inserting parentheses".

But let me check once again if there are other values.

For example, if we have ((2·3)·(4·5)) + 1 = 6·20 + 1 = 121.

Same as original.

Or, (2·(3·4·5)) + 1 = 120 + 1 = 121.

Same.

Or, 2·( (3·4·5) + 1 ) = 122.

Or, 2·(3·(4·5 + 1)) = 126.

Or, 2·(3·(4·(5 + 1))) = 144.

Or, (2·3)·(4·(5 + 1)) = 144.

So, no other values.

Therefore, the answer is 4.

But wait, I remember that sometimes with parentheses, you can get different results.

Wait, let me think of another way.

Suppose we have 2·(3·(4·(5 + 1))) = 144.

Or, 2·( (3·4)·(5 + 1) ) = 144.

Or, (2·3)·( (4·5) + 1 ) = 126.

Or, 2·( (3·(4·5)) + 1 ) = 122.

Or, (2·3·4·5) + 1 = 121.

So, that's five different expressions, but only four different values.

Yes, 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check if there's another value.

Suppose we have (2·(3·(4 + 5))) + 1. But as discussed earlier, this would require adding 4 and 5, which is not allowed.

But if we consider that the original expression is 2·3·4·5 + 1, and we can insert parentheses to do 2·(3·(4 + 5)) + 1, but that would be changing the original expression.

But since the original expression has no addition between 4 and 5, inserting parentheses to do that would be invalid?

Wait, no. Wait, the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

Therefore, inserting parentheses can change the order of operations. For example, if we do 2·(3·(4·(5 + 1))), that is allowed.

But if we do 2·(3·(4 + 5)) + 1, that would be inserting parentheses to do 4 + 5, which is not part of the original expression.

But the original expression has only one addition, between 5 and 1. Therefore, inserting parentheses to do 4 + 5 would require that the addition is between 4 and 5, which is not present.

Therefore, that is not allowed.

Therefore, the value 55 is not allowed.

Therefore, the only values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once again.

Original expression: 2·3·4·5 + 1 = 121.

Other values:

- 2·(3·(4·(5 + 1))) = 144.

- 2·(3·(4·5 + 1)) = 126.

- 2·(3·4·5 + 1) = 122.

- (2·3·4)·(5 + 1) = 144.

- (2·3)·(4·(5 + 1)) = 144.

- 2·( (3·4)·(5 + 1) ) = 144.

- (2·(3·4))·(5 + 1) = 144.

- ( (2·3)·4 )·(5 + 1) = 144.

- 2·( (3·(4·5)) + 1 ) = 122.

- (2·3)·( (4·5) + 1 ) = 126.

- 2·3·( (4·5) + 1 ) = 126.

- (2·3·4·5) + 1 = 121.

So, all these give the same four values.

Therefore, the answer is 4.

But wait, the problem says "how many values can be obtained".

But I need to make sure that there are no other values.

For example, if we have ((2·3)·(4·5)) + 1 = 6·20 + 1 = 121.

Same as original.

Or, ((2·(3·4))·5) + 1 = 24·5 + 1 = 121.

Same.

Or, (2·(3·(4·5))) + 1 = 120 + 1 = 121.

Same.

Therefore, no new values.

Therefore, the answer is 4.

But I need to check if there's a possibility of another value.

Wait, what if we have something like 2·(3 + 4·5) + 1?

But that would be 2·(3 + 20) + 1 = 2·23 + 1 = 46 + 1 = 47.

But again, this requires inserting an addition between 3 and 4·5, which is not part of the original expression.

But the original expression is 2·3·4·5 + 1.

So, inserting parentheses to do 3 + 4·5 would require changing the order of operations, but since the original expression has no addition between 3 and 4, this is not allowed?

Wait, the original expression is 2·3·4·5 + 1. So, the operations are multiplication, multiplication, multiplication, and then addition.

Therefore, inserting parentheses to do 3 + 4·5 would require that the addition is between 3 and 4·5, which is not part of the original expression.

Therefore, this is not allowed.

Therefore, the value 47 is not allowed.

Therefore, the only allowed values are 121, 122, 126, 144.

Therefore, the answer is 4.

But let me check once again.

Is there a way to get 121, 122, 126, 144?

Yes.

Therefore, the total number of different values is 4.

Therefore, the answer is \boxed{4}.

**Final Answer**
\boxed{4}

To determine how many **distinct values** can be obtained from the expression
$$
2 \cdot 3 \cdot 4 \cdot 5 + 1
$$
by **inserting parentheses**, we must consider all valid ways to parenthesize the expression while **preserving the order of the numbers** and **not rearranging them**.

---

### Step 1: Understand the Expression

The original expression is:
$$
2 \cdot 3 \cdot 4 \cdot 5 + 1
$$
According to standard order of operations, multiplication is performed before addition. So the original value is:
$$
(2 \cdot 3 \cdot 4 \cdot 5) + 1 = 120 + 1 = 121
$$

However, by inserting parentheses, we can change the **order of operations**, potentially altering the result.

---

### Step 2: Consider All Valid Parentheses Insertions

We are only allowed to **insert parentheses**, not rearrange the numbers. The expression has:

- Four multiplications: $2 \cdot 3$, $3 \cdot 4$, $4 \cdot 5$
- One addition: $5 + 1$

We can group the multiplications in different ways and apply the addition at different points. The key is to **group the addition with some of the multiplications**.

Let’s explore the different groupings and compute the resulting values:

---

### Step 3: Compute All Possible Values

#### 1. **Original Expression**
$$
(2 \cdot 3 \cdot 4 \cdot 5) + 1 = 120 + 1 = \boxed{121}
$$

#### 2. **Group the addition with 5 and 1, then multiply by 4**
$$
(2 \cdot 3 \cdot 4) \cdot (5 + 1) = 24 \cdot 6 = \boxed{144}
$$

#### 3. **Group the addition with 5 and 1, then multiply by 3**
$$
(2 \cdot 3) \cdot (4 \cdot (5 + 1)) = 6 \cdot 24 = \boxed{144}
$$

#### 4. **Group the addition with 5 and 1, then multiply by 2**
$$
2 \cdot (3 \cdot (4 \cdot (5 + 1))) = 2 \cdot (3 \cdot 24) = 2 \cdot 72 = \boxed{144}
$$

#### 5. **Group the addition with 4 and 5, then multiply by 3 and 2**
$$
2 \cdot (3 \cdot (4 \cdot 5 + 1)) = 2 \cdot (3 \cdot 21) = 2 \cdot 63 = \boxed{126}
$$

#### 6. **Group the addition with 4 and 5, then multiply by 2 and 3**
$$
(2 \cdot 3) \cdot (4 \cdot 5 + 1) = 6 \cdot 21 = \boxed{126}
$$

#### 7. **Group the addition with 3, 4, and 5, then multiply by 2**
$$
2 \cdot (3 \cdot 4 \cdot 5 + 1) = 2 \cdot (60 + 1) = 2 \cdot 61 = \boxed{122}
$$

---

### Step 4: Identify Unique Values

From the above, the **distinct values** obtained are:

- $121$
- $122$
- $126$
- $144$

All other groupings result in one of these values.

---

### Final Answer

$$
\boxed{4}
$$

Solution:
By the associative property of multiplication, it doesn't help to insert parentheses that specify the order of multiplication. For example, the associative property tells us that $(2\cdot(3\cdot 4))\cdot (5+1)$ is the same as $2\cdot3\cdot4\cdot (5+1)$. So the only way to get different values is to group +1 with a different number of factors. We get \begin{align*}
2\cdot 3 \cdot 4 \cdot (5 + 1) &= 144, \\
2\cdot 3 \cdot (4 \cdot 5 + 1) &= 126,\\
2\cdot (3 \cdot 4 \cdot 5 + 1) &= 122, \\
(2\cdot 3 \cdot 4 \cdot 5) + 1 \hphantom{)} &= 121.
\end{align*}In total there are $\boxed{4}$ possible values for the expression.