Question

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

Short 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-by-step solution

Given Information.

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

Explanation.

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

We know that,

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

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.

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.