Question

A classic counting problem is to determine the number of different ways that the letters of "parallel" can be

arranged. Find that number.
The number of different ways that the letters of "parallel" can be arranged is

Answer 1

3,360

Explanation:

There are 8 letters in the word "PARALLEL". Also there are 3 letters L and 2 letters A.

We can arrange 8 letters in 8! different ways. But we need to count how many of them are the sasme arrangments, because latters are not all different.

So, the number of different ways that the letters of "PARALLEL" can be arranged is

`(8!)/(3!\cdot 2!)=(1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8)/(1\cdot 2\cdot 3\cdot 1\cdot 2)=2\cdot 5\cdot 6\cdot 7\cdot 8=60\cdot 56=3,360`

Here we divided by number of different ways to rearrange 3 letters L and by number of different ways to arrange 2 letters E.