# Combination or Permutation?

• May 21st 2009, 02:11 AM
Lupin
Combination or Permutation?
For the problem:
How many ways are there to re-arrange the letters of LITTLE?
Would I use combination, permutation or some other method? And how would I do it?
• May 21st 2009, 02:30 AM
Arrangements with repeated letters
Hello Lupin
Quote:

Originally Posted by Lupin
For the problem:
How many ways are there to re-arrange the letters of LITTLE?
Would I use combination, permutation or some other method? And how would I do it?

It's basically a permutation problem, but with repeated letters you need to do a bit of extra work. You do it like this:

If all 6 letters were different - call them $L_1, I, T_1, T_2, L_2, E$ - then there would be $6!$ arrangements.

But the $L_1$ and $L_2$ are indistinguishable. And they can be arranged among themselves in $2! = 2$ ways. So our $6!$ arrangements will contain each of these $2!$ arrangements as if they were different. And, of course, they're not. So we need to divide by $2!$.

And the same goes for $T_1$ and $T_2$: we need to divide by $2!$ again, to get rid of the duplication caused by the $T$'s.

So the final answer is that there are $\frac{6!}{2!2!} = 180$ different arrangements.

You can generalise this and say that if there are $n$ items, with $a$ items repeated of the first kind, $b$ repeated of the second kind, and so on, the number of different arrangements is:

$\frac{n!}{a!b!...}$