Question

How many three letter arrangements can be formed if a letter is used only once? [TIGER]

Answer 1

`60`

Explanation:

This is equivalent to asking in how many different ways can u select 3 from an available 5 and arrange them.

This is then the permutation `""^5P_3=(5!)/((5-3)!)=60`.

Answer 2

`60`

Explanation:

There are `5` ways to choose the first letter, `4` to choose the next letter and `3` ways to choose the third letter, hence `5xx4xx3 = 60` ways to choose an arrangement of `3` letters from `5`.

In general, if you have `n` distinct objects from which to choose an arrangement of `k` items then the number of ways you can do it is:

`""^nP_k = (n!)/((n-k)!)`

In our example, `n=5`, `k=3` and:

`""^5P_3 = (5!)/((5-3)!) = (5!)/(2!) = (5xx4xx3xxcolor(red)(cancel(color(black)(2)))xxcolor(red)(cancel(color(black)(1))))/(color(red)(cancel(color(black)(2)))xxcolor(red)(cancel(color(black)(1)))) = 5xx4xx3=60`

If the order of the chosen items does not matter, then the number of ways to choose `k` items from `n` is:

`""^nC_k = ((n),(k)) = (n!)/(k!(n-k)!)`