Q27. - Algebra 2

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 parallel.

Step-by-Step Solution

Answer

The situation is a permutation and the number of possibilities is 3360. 

Step 1. Given Information.

Given to arrange the letters in the word parallel. 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 8. The letter a is repeated twice and the letter l is repeated thrice.

Plugging the values:

 $\begin{array}{l}P=\frac{8!}{2!3!}\\ P=\frac{8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3!}{2\cdot 1\cdot 3!}\\ P=\frac{8\cdot 7\cdot 6\cdot 5\cdot 4}{2\cdot 1}\\ P=\frac{6720}{2}\\ P=3360\end{array}$

Step 3. Conclusion .

Hence, the given situation is a permutation with 3360 possibilities.