Q9. - Algebra 2

Question

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

 

Seven shoppers in line at a checkout counter.

Step-by-Step Solution

Answer

The situation is a permutation and the number of possibilities is 5040. 

Step 1. Given Information.

Given to choose seven shoppers in line at a checkout counter. 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 shoppers are to be arranged i.e., the given situation is a permutation.

 

Here n=7,r=7

Plugging the values:

 $\begin{array}{l}P\left(7,7\right)=\frac{7!}{\left(7-7\right)!}\\ P\left(7,7\right)=\frac{7!}{0!}\\ P\left(7,7\right)=\frac{7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2\cdot 1}{1}\\ P\left(7,7\right)=5040\end{array}$

Step 3. Conclusion .

Hence, the given situation is a permutation with 5040 possibilities.