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

Jan 16, 2016

$60$

#### 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.

Jan 16, 2016

$60$

#### 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 $5 \times 4 \times 3 = 60$ 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)!)