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Chapter 34: Problem 1081

How many committees of four members each can be formed from a group of seven persons?

Short Answer

Expert verified
In this problem, we use the concept of combinations to calculate the number of committees of four members that can be formed from a group of seven people. By applying the formula \(\binom{n}{r}=\frac{n!}{r!(n-r)!}\), where n = 7 and r = 4, we compute \(\binom{7}{4}=\frac{7!}{4!(7-4)!}\) or \(\binom{7}{4}=\frac{5,040}{24 脳 6}\), which results in 35 different possible committees.

Step by step solution

01

Understanding the concept of combinations

The concept of combinations refers to the selection of items from a larger set, where the order of the items does not matter. It is represented by the symbol C(n, r) or \(\binom{n}{r}\), where n is the total number of items in the set and r is the number of items to be selected.
02

Identify the values of n and r in the given problem

In this problem, there are seven people, which makes n = 7. We want to form a committee of four members, which means r = 4. So, we will calculate the combination C(7, 4) or \(\binom{7}{4}\).
03

Apply the combination formula

The formula for combinations is given by: \(\binom{n}{r}=\frac{n!}{r!(n-r)!}\), where n! denotes the factorial of n (the product of all positive integers up to n).
04

Calculate the factorials

First, we need to calculate the factorials of n, r, and (n-r): - 7! = 7 脳 6 脳 5 脳 4 脳 3 脳 2 脳 1 = 5,040 - 4! = 4 脳 3 脳 2 脳 1 = 24 - (7-4)! = 3! = 3 脳 2 脳 1 = 6
05

Calculate the combination

Now we can plug the factorials into the formula: \(\binom{7}{4}=\frac{7!}{4!(7-4)!}=\frac{5,040}{24 脳 6}=\frac{5,040}{144}=35\) Therefore, the number of committees of four members that can be formed from a group of seven people is 35.

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