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Chapter 4: Problem 93

In how many ways can a sample (without replacement) of 5 items be selected from a population of 15 items?

Short Answer

Expert verified
The number of ways a sample of 5 items can be selected from a population of 15 items is 3,003 ways.

Step by step solution

01

Understand the problem

In this exercise, a sample of 5 items needs to be selected from a population of 15 items without replacement, meaning the same item cannot be chosen twice. The problem is asking for the number of ways this can be done.
02

Recall the formula for combination

The formula for a combination is given by \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and ! indicates factorial.
03

Plug in the values into the formula

For this problem, n equals to 15 (the total number of items), and k equals to 5 (the number of items to choose). So, \(C(15, 5) = \frac{15!}{5!(15-5)!}\).
04

Calculate the factorial of 15, 5, and 10

Calculate the factorial for 15, 5, and 10. 15! equals to 1,307,674,368,000, 5! equals to 120 and 10! equals to 3,628,800.
05

Substitute the factorials into the formula and simplify

Substitute the calculated factorials back into the formula and simplify. \(C(15, 5) = \frac{1,307,674,368,000}{120 * 3,628,800}\). After performing the division, the solution is 3,003.

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