Chapter 10.5, Problem 38E

To determine

Tocalculate:

Total ways that $3$ roles filled.

Answer to Problem 38E

Total $720$ ways that $3$ roles can be filled.

Explanation of Solution

Given information:

Total students: $10$

Diffreent roles: $3$

Formula used:

Expression for permutation

  ${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Where,

  $n=$ objects

  $r=$ time

Calculation:

There are $10$ students ausitioning for $3$ diffreeny roles. The order of selecting candidates for roles dows matter.

If there are three roles king, queen and minister, then it is required to know number of permutations of $3$ roles chosen from $10$ students.

So, now apply the combination. The expression of permutation is as follows:

  ${}_n{P}_r=\frac{n!}{\left(n-r\right)!}$

Therefore the permutation is as follows:

  $\begin{array}{l}{}_{10}{P}_3=\frac{10!}{\left(10-3\right)!}\\ =\frac{10!}{7!}\\ =\frac{10\cdot 9\cdot 8\cdot 7!}{7!}\\ =720\end{array}$

Thus, the value of ${}_{10}{P}_3$ is $720$ .

Hence, there are $720$ ways that $3$ roles can be filled.