In how many ways can 4 different books be selected from a shelf of 10 books?

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Question: In how many ways can 4 different books be selected from a shelf of 10 books?

Options:

Correct Answer: 210

Solution:

The number of ways to choose 4 books from 10 is C(10,4) = 210.

In how many ways can 4 different books be selected from a shelf of 10 books?

In how many ways can 4 different books be selected from a shelf of 10 books?

Step 1: Understand that we need to choose 4 books from a total of 10 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 = 10 (total books) and r = 4 (books to choose).
Step 5: Plug the values into the formula: C(10, 4) = 10! / (4! * (10 - 4)!)
Step 6: Simplify the equation: C(10, 4) = 10! / (4! * 6!)
Step 7: Calculate 10! = 10 × 9 × 8 × 7 × 6! (we can cancel 6! in the numerator and denominator).
Step 8: Now we have C(10, 4) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 9: Calculate the numerator: 10 × 9 × 8 × 7 = 5040.
Step 10: Calculate the denominator: 4 × 3 × 2 × 1 = 24.
Step 11: Divide the numerator by the denominator: 5040 / 24 = 210.
Step 12: Therefore, the number of ways to choose 4 books from 10 is 210.

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.