Question

How many different arrangements of the letters of the Hawaiian word aloha are possible?

Short answer

The number of different possible arrangements is120.

Step-by-step solution

Step 1. Given Information.

Given, to arrange the letters in the word aloha. The number of different possible arrangements is to be determined.

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.

Here, n=5,r=5

Plugging the values:

$\begin{array}{l}P\left(5,5\right)=\frac{5!}{\left(5-5\right)!}\\ P\left(5,5\right)=\frac{5!}{0!}\\ P\left(5,5\right)=\frac{5\cdot 4\cdot 3\cdot 2\cdot 1}{1}\\ P\left(5,5\right)=120\end{array}$

Step 3. Conclusion.

Hence, the number of different possible arrangements is 120.