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

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

Step 1: Understand that we want to choose 5 books from a total of 10 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 = 10 (total books) and r = 5 (books to choose).
Step 5: Plug the values into the formula: C(10, 5) = 10! / (5! * (10 - 5)!)
Step 6: Simplify the equation: C(10, 5) = 10! / (5! * 5!)
Step 7: Calculate 10! = 10 × 9 × 8 × 7 × 6 × 5! (we can cancel 5! in the numerator and denominator)
Step 8: Now we have C(10, 5) = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1)
Step 9: Calculate the numerator: 10 × 9 × 8 × 7 × 6 = 30240
Step 10: Calculate the denominator: 5 × 4 × 3 × 2 × 1 = 120
Step 11: Divide the numerator by the denominator: 30240 / 120 = 252.
Step 12: Therefore, there are 252 ways to choose 5 different books from a shelf of 10.

Combinatorics – The question 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.