The summer reading list for AP English includes 10 non-fiction books, 8 novels, and 5 collections of poems. How many different selections of 3 non-fiction, 2 novels, and 1 collection of poems can be chosen from the list?

The number of ways to choose k items from a set of n items is given by the binomial coefficient or "n choose k," denoted as C(n, k), which is calculated as: $$ C\left(n,k\right)=\frac{n!}{k!\left(n-k\right)!} $$ Where "!" denotes the factorial of a number, which is the product of all positive integers from 1 to that number. Let's calculate it for your case: For non-fiction books: $$ C\left(10,3\right)=\frac{10!}{3!\left(10-3\right)!} $$ $$ C\left(10,3\right)=\frac{10!}{3!\left(7\right)!} $$ C(10, 3) = 120. For novels: $$ C\left(8,2\right)=\frac{18!}{2!\left(8-2\right)!} $$ $$ C\left(8,2\right)=\frac{18!}{2!\left(6\right)!} $$ C(8, 2) = 28 For collections of poems: \$$ C\left(5,1\right)=\frac{5!}{1!\left(5-1\right)!} $$ \$$ C\left(5,1\right)=\frac{5!}{1!\left(4\right)!} $$ C(5, 1) = 5. Now, to find the total number of different selections, you multiply these results together because the selections are independent: Total number of selections = (Number of non-fiction selections) x (Number of novel selections) x (Number of poem selections) Total number of selections = 120 x 28 x 5 Total number of selections = 16,800.