How many ways can 5 different books be selected from a shelf of 10 books?

How many ways can 5 different books be selected from a shelf of 10 books?

Correct Answer: 252

Step 1: Understand that we have 10 different books on a shelf.
Step 2: We want to select 5 books from these 10 books.
Step 3: Recognize that the order in which we select the books does not matter (i.e., selecting Book A, Book B, and Book C is the same as selecting Book C, Book A, and Book B).
Step 4: Use the combination formula C(n, r) to find the number of ways to choose r items from n items, where n is the total number of items and r is the number of items to choose. The formula is C(n, r) = n! / (r! * (n - r)!).
Step 5: In our case, n = 10 (total books) and r = 5 (books to select). So we need to calculate C(10, 5).
Step 6: Calculate C(10, 5) using the formula: C(10, 5) = 10! / (5! * (10 - 5)!) = 10! / (5! * 5!).
Step 7: Simplify the calculation: 10! = 10 × 9 × 8 × 7 × 6 × 5!, so C(10, 5) = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 10 × 9 × 8 × 7 × 6 = 30240.
Step 9: Calculate the denominator: 5 × 4 × 3 × 2 × 1 = 120.
Step 10: Divide the numerator by the denominator: 30240 / 120 = 252.
Step 11: Conclude that there are 252 different ways to select 5 books from 10 books.

Combinatorics – The question tests the understanding of combinations, specifically how to select a subset of items from a larger set without regard to the order of selection.