How many ways can 5 different books be chosen from a shelf of 15 books?

How many ways can 5 different books be chosen from a shelf of 15 books?

Correct Answer: 3003

Step 1: Understand that we need to choose 5 books from a total of 15 books.
Step 2: Recognize that the order in which we choose the books does not matter. This means we will use combinations, not permutations.
Step 3: The formula for combinations is C(n, r) = n! / (r! * (n - r)!), where n is the total number of items, r is the number of items to choose, and '!' denotes factorial.
Step 4: In our case, n = 15 (total books) and r = 5 (books to choose).
Step 5: Plug the values into the formula: C(15, 5) = 15! / (5! * (15 - 5)!) = 15! / (5! * 10!).
Step 6: Calculate 15! / (5! * 10!) using the factorial values: 15! = 15 × 14 × 13 × 12 × 11 × 10!, so we can cancel 10! in the numerator and denominator.
Step 7: This simplifies to (15 × 14 × 13 × 12 × 11) / (5 × 4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 15 × 14 × 13 × 12 × 11 = 360360.
Step 9: Calculate the denominator: 5 × 4 × 3 × 2 × 1 = 120.
Step 10: Divide the numerator by the denominator: 360360 / 120 = 3003.
Step 11: Therefore, the number of ways to choose 5 books from 15 is 3003.

Combinatorics – The problem tests the understanding of combinations, specifically how to calculate the number of ways to choose a subset of items from a larger set without regard to the order of selection.