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Chapter 1: Problem 27

If 12 people are to be divided into 3 committees of respective sizes \(3,4,\) and \(5,\) how many divisions are possible?

Short Answer

Expert verified
There are \(\underline{27,720}\) possible divisions of the 12 people into three committees of respective sizes 3, 4, and 5.

Step by step solution

01

Calculate the number of ways to form the first committee of 3 people

To choose the 3 people for the first committee, we have 12 people to choose from and need to select 3. In combinatorics, we can use the combination formula to calculate this: \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) Where \(n\) is the total number of items, \(k\) is the number of items to choose from the total, and \(n!\) denotes the factorial of \(n\). Using the combination formula, we have: \(\binom{12}{3} = \frac{12!}{3!(12-3)!} = \frac{12!}{3!9!} = 220\) Thus, there are 220 ways to form the first committee of 3 people.
02

Calculate the number of ways to form the second committee of 4 people

Next, we need to choose the 4 people for the second committee. Since we have already chosen 3 people for the first committee, we now have 9 remaining people to choose from (12 - 3 = 9). We will use the combination formula again: \(\binom{9}{4} = \frac{9!}{4!(9-4)!} = \frac{9!}{4!5!} = 126\) Thus, there are 126 ways to form the second committee of 4 people.
03

Calculate the number of ways to form the third committee of 5 people

Finally, we need to select the remaining 5 people for the third committee. Since we have already chosen 3 people for the first committee and 4 people for the second committee, we have 5 remaining people to choose from (12 - 3 - 4 = 5). This means that the final committee is automatically formed by the remaining 5 people: \(\binom{5}{5} = \frac{5!}{5!(5-5)!} = \frac{5!}{5!0!} = 1\) (Note: 0! = 1) Thus, there is only 1 way to form the third committee of 5 people.
04

Multiply the number of ways to form each committee to find the total number of divisions

We have found the number of ways to form each committee separately. To find the total number of divisions, we need to multiply the number of ways to form each committee: Total divisions = (Number of ways to form the first committee) 脳 (Number of ways to form the second committee) 脳 (Number of ways to form the third committee) Total divisions = 220 脳 126 脳 1 = 27720 Thus, there are 27,720 possible divisions of the 12 people into the three committees of respective sizes 3, 4, and 5.

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