Q29. - Algebra 2

Question

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

 

Selecting three of fifteen flavors of ice cream at the grocery store.

Step-by-Step Solution

Answer

The situation is a combination and the number of possibilities is 455. 

Step 1. Given Information.

Given to select three of fifteen flavors of ice cream at the grocery store. 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 flavors are just picked and not arranged i.e. the given situation is a combination.

 

Here n=15,r=3

Plugging the values:

 $\begin{array}{l}C\left(15,3\right)=\frac{15!}{\left(15-3\right)!3!}\\ C\left(15,3\right)=\frac{15!}{12!3!}\\ C\left(15,3\right)=\frac{15\cdot 14\cdot 13\cdot 12!}{12!\cdot 3\cdot 2\cdot 1}\\ C\left(15,3\right)=\frac{2730}{6}\\ C\left(15,3\right)=455\end{array}$

Step 3. Conclusion .

Hence, the given situation is a combination with 455 possibilities.