Chapter 15.6, Problem 24WE

To determine

To find: The number of ways in which the $3$ red, $4$ blue, and $2$ green pens can be distributed to $9$ students seated in a row.

Answer to Problem 24WE

Thenumber of ways in which the distribution can be made is $1260$ .

Explanation of Solution

Given:

The variety of pens are $3$ red, $4$ blue, and $2$ green pens.

The total students are 9.

Calculation:

Consider that there $3$ red, $4$ blue, and $2$ green pens and total of 9 students.

Consider that for a set of $n$ elements that has $n_1$ element of one kind and $n_2$ element of the other kind and so on then the number of permutation of $n$ object taken $n$ at a time is,

  $P=\frac{n!}{n_1!n_2!\mathrm{....}}$

Then, the number of ways of in which the pens are distribution is,

  $\begin{array}{c}P=\frac{9!}{3!4!2!}\\ =\frac{9\times 8\times 7\times 6\times 5\times 4\times 3\times 2\times 1}{\left(3\times 2\times 1\right)\left(4\times 3\times 2\times 1\right)\left(2\times 1\right)}\\ =1260\end{array}$

Thus, the number of distribution that can be made is $1260$ .