Chapter 11.1, Problem 29PPSE

Solution

To calculate: The number of ways in which 10 time slots can be assigned if fifteen students ask to visit a college admissions counselor and each schedule visit included one student.

The resultantanswer is $10,897,286,400$ ways.

Given information:

Fifteen students ask to visit a college admissions counselor and each schedule visit includes one student.

Formula used:

If $P{}_r$ represents the number of permutations of n items taken r at a time, then $P{}_r=\frac{n!}{\left(n-r\right)!}$ .

Where,

  $\begin{array}{l}n=\text{Number of trials}\\ r=\text{Number of successes}\end{array}$

Calculation:

The total number of student is 15. Each schedule visit included one student and 10 time slot are to be assigned.

Order matters here, therefore use permutation. Then the expression becomes: ${}_{15}{P}_{10}$

Rewrite the expression using the formula $P{}_r=\frac{n!}{\left(n-r\right)!}$ ,

  $\begin{array}{c}{}_{15}{P}_{10}=\frac{15!}{\left(15-10\right)!}\\ =\frac{15!}{5!}\end{array}$

Now expand the factorial,

  $\begin{array}{c}\frac{15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5!}{5!}=15\cdot 14\cdot 13\cdot 12\cdot 11\cdot 10\cdot 9\cdot 8\cdot 7\cdot 6\\ =10,897,286,400\end{array}$

Therefore, there are $10,897,286,400$ ways in which 10 time slots should be assigned.