Question

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

An arrangement of the letters in the word intercept

Short answer

The situation is apermutationand the number of possibilitiesis90720.

Step-by-step solution

Step 1. Given Information.

Given to arrange the letters in the word intercept. 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 a letter does affect the final outcome i.e., every arrangement is different. Hence the letters are to be arranged i.e., the given situation is a permutation.

The number of permutations of n objects of which p are alike and q are alike is $\frac{n!}{p!q!}$

The number of letters in the word is 9 and letters e and t are repeated twice each.

Plugging the values:

$\begin{array}{l}P=\frac{9!}{2!2!}\\ P=\frac{9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot 2!}{2\cdot 1\cdot 2!}\\ P=\frac{181440}{2}\\ P=90720\end{array}$

Step 3. Conclusion.

Hence, the given situation is a permutation with 90720 possibilities.