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

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

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:

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

Hence, the number of different possible arrangements is 120.