How many ways can 3 different books be chosen from a set of 7?

How many ways can 3 different books be chosen from a set of 7?

Step 1: Understand that we want to choose 3 books from a total of 7 books.
Step 2: Recognize that the order in which we choose the books does not matter, so we will use combinations.
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 this case, n = 7 (total books) and r = 3 (books to choose).
Step 5: Plug the values into the formula: C(7, 3) = 7! / (3! * (7 - 3)!)
Step 6: Calculate 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040.
Step 7: Calculate 3! = 3 × 2 × 1 = 6.
Step 8: Calculate (7 - 3)! = 4! = 4 × 3 × 2 × 1 = 24.
Step 9: Now substitute these values back into the formula: C(7, 3) = 5040 / (6 * 24).
Step 10: Calculate 6 * 24 = 144.
Step 11: Finally, divide 5040 by 144 to get C(7, 3) = 35.

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.