How many ways can 4 different books be chosen from a shelf of 10 books?

How many ways can 4 different books be chosen from a shelf of 10 books?

Correct Answer: 210

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)!) = 10! / (4! * 6!).
Step 6: Calculate 10! = 10 × 9 × 8 × 7 × 6! (we can cancel 6! in the numerator and denominator).
Step 7: Now we have C(10, 4) = (10 × 9 × 8 × 7) / (4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 10 × 9 = 90, then 90 × 8 = 720, and finally 720 × 7 = 5040.
Step 9: Calculate the denominator: 4 × 3 = 12, then 12 × 2 = 24, and finally 24 × 1 = 24.
Step 10: Now divide the numerator by the denominator: 5040 / 24 = 210.
Step 11: Therefore, the number of ways to choose 4 books from 10 is 210.

Combinations – The concept of combinations is used to determine the number of ways to choose a subset of items from a larger set, where the order of selection does not matter.