Question

Determine whether each situation involves a permutation or a combination. Then find the number of possibilities.

Placing an algebra book, a geometry book, a chemistry book, an English book, and a health book on a shelf.

Short answer

The situation is apermutationand the number of possibilities is120.

Step-by-step solution

Step 1. Given Information.

Given to place an algebra book, a geometry book, a chemistry book, an English book, and a health book on a shelf. It is to be determined if the situation involves a permutation or a combination and then the number of possibilities are to be calculated.

Step 2. Calculation.

A permutation is when n objects are available and r are to be picked and arranged in a certain order and the number of permutations is given by $P\left(n,r\right)=\frac{n!}{\left(n-r\right)!}$

A combination is when n objects are available and r are to be picked without arrangement and the number of combinations is given by $C\left(n,r\right)=\frac{n!}{\left(n-r\right)!r!}$

Here, the order of choosing does affect the final outcome i.e., every arrangement is different. Hence the books are to be arranged i.e., the given situation is a permutation.

Here n=5,r=5

Plugging the values:

$\begin{array}{l}P\left(5,5\right)=\frac{5!}{\left(5-5\right)!}\\ P\left(5,5\right)=\frac{5!}{0!}\\ P\left(5,5\right)=\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{1}\\ P\left(5,5\right)=5\cdot 4\cdot 3\cdot 2\cdot 1\\ P\left(5,5\right)=120\end{array}$

Step 3. Conclusion.

Hence, the given situation is a permutation with 120 possibilities.