In how many ways can 5 different items be selected from 10 items?

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Question: In how many ways can 5 different items be selected from 10 items?

Options:

Correct Answer: 252

Solution:

The number of ways to select 5 items from 10 is C(10, 5) = 252.

In how many ways can 5 different items be selected from 10 items?

In how many ways can 5 different items be selected from 10 items?

Step 1: Understand that we want to select 5 different items from a total of 10 items.
Step 2: Recognize that the order in which we select the items 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 select, and '!' denotes factorial.
Step 4: In our case, n = 10 (total items) and r = 5 (items to select).
Step 5: Plug the values into the formula: C(10, 5) = 10! / (5! * (10 - 5)!) = 10! / (5! * 5!).
Step 6: Calculate 10! = 10 × 9 × 8 × 7 × 6 × 5! (we can cancel 5! in the numerator and denominator).
Step 7: Now we have C(10, 5) = (10 × 9 × 8 × 7 × 6) / (5 × 4 × 3 × 2 × 1).
Step 8: Calculate the numerator: 10 × 9 × 8 × 7 × 6 = 30240.
Step 9: Calculate the denominator: 5 × 4 × 3 × 2 × 1 = 120.
Step 10: Divide the numerator by the denominator: 30240 / 120 = 252.
Step 11: Therefore, the number of ways to select 5 items from 10 is 252.

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.