1. ## 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?

2. ## Arrangements with repeated letters

Hello Lupin
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 $\displaystyle L_1, I, T_1, T_2, L_2, E$ - then there would be $\displaystyle 6!$ arrangements.

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

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

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

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

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