Question

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

Selecting nine books to check out of the library from a reading list of twelve.

Short answer

The situation is acombinationand the number of possibilitiesis220.

Step-by-step solution

Step 1. Given Information.

Given to select nine books to check out of the library from a reading list of twelve. 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 doesn’t affect the final outcome. Hence the books are just picked and not arranged i.e. the given situation is a combination.

Here n=12,r=9

Plugging the values:

$\begin{array}{l}C\left(12,9\right)=\frac{12!}{\left(12-9\right)!9!}\\ C\left(12,9\right)=\frac{12!}{3!9!}\\ C\left(12,9\right)=\frac{12\cdot 11\cdot 10\cdot 9!}{3\cdot 2\cdot 1\cdot 9!}\\ C\left(12,9\right)=\frac{1320}{6}\\ C\left(12,9\right)=220\end{array}$

Step 3. Conclusion.

Hence, the given situation is a combination with 220 possibilities.