How many ways can 3 books be selected from a shelf of 10 books? (2022)

How many ways can 3 books be selected from a shelf of 10 books? (2022)

Step 1: Understand that we want to choose 3 books from a total of 10 books.
Step 2: Recognize that the order in which we select the books does not matter. This means we will use combinations, not permutations.
Step 3: The formula for combinations is given by nCr = 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 = 3 (books to choose).
Step 5: Calculate 10C3 using the formula: 10C3 = 10! / (3! * (10 - 3)!)
Step 6: Simplify the calculation: 10C3 = 10! / (3! * 7!) = (10 * 9 * 8) / (3 * 2 * 1)
Step 7: Calculate the numerator: 10 * 9 * 8 = 720.
Step 8: Calculate the denominator: 3 * 2 * 1 = 6.
Step 9: Divide the numerator by the denominator: 720 / 6 = 120.
Step 10: Conclude that there are 120 ways to select 3 books from 10.

Combinatorics – The study of counting, specifically how to choose items from a larger set without regard to the order of selection.
Binomial Coefficient – The formula used to calculate the number of ways to choose k items from n items, denoted as nCk or C(n, k).