Question

How many different ways can you arrange of the letter of hawaiian word oloha are possible?

Answer 1

oloha
a, h, l, o, o
So, to find the amount of ways we can arrange these letters without accounting for repeats, we will take 5!. This equals 120.

However, we have two letters that are the same. So, for any position we place one 'o', we can place the other 'o' in the same place and still have the same arrangement. This invalidates half of our possible outcomes. (Like, for an arrangement of "aohlo", there will be another arrangement that is the same because we switch the o's around.)

Sorry, its pretty hard to explain well for something that has 120 arrangements. Maybe the concept would make more sense to try this on a shorter word like "mom"
==> 3! = 6 arrangements
mmo mmo
mom mom
omm omm

Answer 2

The way I would think about this problem is like so:

First, since you have two o's, let's place those first. This means, you have a total of
5-choose-2 ways to put the o's. Then you have three places remaining, with three different letters, so that 3! ways to arrange those. Thus, your answer is \[\left( \begin{matrix} 5 \\ 2 \end{matrix} \right) * 3! = 10*6=60\]

Answer 3

thanks