Q.1.27 - A First Course in Probability

Question

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

Step-by-Step Solution

Answer

$27720$Use of the multinomials yields: there are $\left(\begin{array}{c}12\\ 3,4,5\end{array}\right)=27720$ways in which$12$ people can be divided into three committees $3,4$and $5$members.

Step 1 Given Information.

Let $12$people are to be divided into $3$committees of respective sizes $3,4,$and$5$

Step 2 Explanation.

For division of $n$different objects into groups of fixed sizes$\left(n_1,n_2,\dots n_k\right)$.

We know that, 

$\left(\begin{array}{c}n\\ n_1,n_2,\dots ,n_k\end{array}\right)=\frac{n!}{n_1!n_2!\dots n_k!}, n_1+n_2+\dots +n_k=n$

Step 3 Explanation.

$n=12$

$n_1=3$

$n_2=4$

$n_3=5$

$k=3$

$\left(\begin{array}{c}12\\ 3,4,5\end{array}\right)=\frac{12}{3!4!5!}=27720$

There are$\left(\begin{array}{c}12\\ 3,4,5\end{array}\right)=27720$ ways in which $12$people can be divided into three committees $3,4$and $5$members.

Step 4 Explanation.

$27720$Use of the multinomials yields: there are $\left(\begin{array}{c}12\\ 3,4,5\end{array}\right)=27720$ways in which$12$ people can be divided into three committees$3,4$ and $5$members.