# How many words can be formed by taking four letters at a time out of the letters of the word MATHEMATICS? [duplicate]

The question is:

How many words can be formed by taking four letters at a time out of the letters of the word MATHEMATICS?

I know how to do it for non-repeating letters. You simply permute $$4$$ things taken from $$11$$ at a time. But how do I do this? How to subtract out the words that have repeated letters in them? What should I divide the total permutations with?

## 1 Answer

I think the easiest way is to separate out cases according to how many repeated letters there are in the word you make. In this case there are only three possibilities: no repeats, one letter occurs twice, two letters occur twice each.

The first case is easy since there are now eight distinct letters and you need to take four of them, in order.

For the second case, how many ways are there to choose the repeated letter? How many ways are there to position two copies of the repeated letter? How many ways are there to fill the other two positions (in order)?

The third case is similar to the second.