[["Problem: How many different triangles can be formed using three vertices of a hexagon as vertices of a triangle? [asy]size(75);\ndraw(dir(0)--dir(30)--dir(110)--dir(175)--dir(250)--dir(300)--cycle);\n[/asy]\nAnswer:",["\n"]],["Problem: A standard deck of cards has 26 cards which are considered \"red\" (the 'hearts' and 'diamonds') and 26 which are considered \"black\" (the 'spades' and 'clubs'). In how many different ways can we choose two red cards from the deck? (Note: Order matters in the sense that drawing an ace of hearts followed by jack of diamonds is different than drawing a jack of diamonds followed by ace of hearts, for instance.)\nAnswer:",["\n"]],["Problem: Each square of the three by three grid is painted so that the whole picture has at least the two lines of symmetry indicated. Each grid square is painted one solid color. What is the maximum number of colors that could have been used? [asy]size(100);\ndraw((0,0)--(0,3)--(3,3)--(3,0)--cycle);\ndraw((1,0)--(1,3));\ndraw((2,0)--(2,3));\ndraw((0,1)--(3,1));\ndraw((0,2)--(3,2));\ndraw((-0.2,-0.2)--(3.2,3.2),dashed);\ndot((0,0));dot((3,3)); dot((1.5,0)); dot((1.5,3));\ndraw((1.5,-0.3)--(1.5,3.3), dashed);\n[/asy]\nAnswer:",["\n"]],["Problem: A bag contains two red beads and two green beads. You reach into the bag and pull out a bead, replacing it with a red bead regardless of the color you pulled out. What is the probability that all beads in the bag are red after three such replacements? Express your answer as a common fraction.\nAnswer:",["\n"]],["Problem: Ten circles are all the same size. Each pair of these circles overlap but no circle is exactly on top of another circle. What is the greatest possible total number of intersection points of these ten circles?\nAnswer:",["\n"]],["Problem: I want to read 4 books over the next month. My bookshelf has 12 different books. In how many ways can I choose which books to read over the next month, without regard to the order that I read them?\nAnswer:",["\n"]],["Problem: How many primes are in the row of Pascal's Triangle that starts with a $1$ followed by a $6$?\nAnswer:",["\n"]],["Problem: Suppose that we win $\\$3$ if we flip a heads on a coin toss, but lose $\\$2$ if we flip tails. What is the expected value, in dollars, of our winnings after one flip?\nAnswer:",["\n"]],["Problem: A Senate committee consists of 5 Republicans, 6 Democrats, and 2 Independents. A subcommittee of 3 members is randomly chosen. What is the probability that the subcommittee consists of 1 Republican, 1 Democrat, and 1 Independent?\nAnswer:",["\n"]],["Problem: For how many ordered pairs of positive integers $(x,y)$ is $x+2y =\n100$?\nAnswer:",["\n"]]]
