Question

How many three letter arrangements of the word ALGORA are there?

Answer 1

72 arrangements

Explanation:

This is a permutation question in that we care about the order of the letters. The general formula is:

`P_(n,k)=(n!)/((n-k)!); n="population", k="picks"`

If all the letters were different, we'd have:

`P_(6,3)=(6!)/(3!)=720/6=120`

However, we have to deal with the repeated A.

Unfortunately, we can't simply divide by `2!` to get rid of the double A because not all our permutations are going to have it. And so we have to deal with the cases separately - one where at most one A is selected, and the other with both As.

At most one A

That's 5 distinct letters and we're selecting 3 of them:

`P_(5,3)=(5!)/(2!)=120/2=60`

Both As

There are 4 letters that are not A and it can be placed in any one of 3 places (first letter, second, or third). That gives:

`4xx3=12`

We now add these results:

`60+12=72`