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Chapter 11: Problem 45

From a club of 20 people, in how many ways can a group of three members be selected to attend a conference?

Short Answer

Expert verified
You can select three members from a group of 20 in 1140 different ways.

Step by step solution

01

Understanding Combination Formula

Combination formula, often denoted \(C(n,r)\) or \(nCr\), is given by \[C(n,r) = \frac{n!}{r!(n-r)!}\] where 'n' is the total number of items, 'r' is the number of items to choose, and '!' denotes factorial, which is the product of all positive integers up to that number.
02

Plug the numbers into the formula

Here the value of 'n' is 20 (total people in the club) and 'r' is 3 (number of people to select). Plugging these values in combination formula gives \[C(20,3) = \frac{20!}{3!(20-3)!}\]
03

Simplify the formula

First calculate the values of factorials, 20!, 3!, and 17!. Now simplify the formula: \[C(20,3) = \frac{20*19*18*17!}{3*2*1*17!}\] The term '17!' gets canceled out from numerator and denominator. After this, \(C(20,3)\) simplifies to \(20*19*18 / (3*2*1)\)
04

Compute the final answer

Now compute the division and multiplication to get the final answer: \[C(20,3) = 20*19*18 / 6 = 1140\] So, there are 1140 ways of selecting three members from a group of 20.

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