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

Question

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?

You have to divide the total permutation with the $n!$ where $n$ is the number of repeating words that are appearing in the word — Vedant Chourey, Jul 24 '19 at 09:30
Possible abstract duplicates : https://math.stackexchange.com/questions/2115072/how-many-3-letter-words-can-be-formed-from-the-word-testbook, https://math.stackexchange.com/questions/2066257/4-letter-words-taken-from-the-letters-of-concentrations, https://math.stackexchange.com/questions/20238/6-letter-permutations-in-mississippi and its linked questions... — Arnaud D., Jul 24 '19 at 09:37
Possible concrete duplicate of How many perms of 4 from letters of MATHEMATICS? — Arnaud D., Jul 24 '19 at 09:38
This seems to be the most general formulation of the problem. — Arnaud D., Jul 24 '19 at 09:42

score 1 · Answer 1 · answered Jul 24 '19 at 09:29

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.

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

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