How many ways are there to divide 12 people into a group of 3, a group of 4, and a group of 5, if Henry has to be in the group of 4?
I need to count the number of ways to form the groups, given the constraint that Henry has to be in the group of 4. I can start by choosing the other 3 people in Henry's group, which can be done in 11 choose 3 ways, since there are 11 people left to choose from. Now I have 8 people left to divide into a group of 3 and a group of 5. I can either choose the group of 3 first, or the group of 5 first, but either way, I get the same number of ways, which is 8 choose 3, or 8 choose 5. So the total number of ways to form the groups is (11 choose 3) times (8 choose 3). I can use the formula for binomial coefficients to calculate these numbers. 11 choose 3 is 11! / (3! * 8!), which is 165. 8 choose 3 is 8! / (3! * 5!), which is 56. So the final answer is 165 times 56, which is 9240. 