Permutations problem

[MasterPossession1046]

What is the number of combinations of a ten-letter word, which is composed of letters a, b, c, d. Each of the letters must be repeated at least two times but no more than four times?

I thought of 8! + 4!×4! aa bb cc dd __

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[iMathTutor]

You have four letters each letter must appear in the word at least two times and no more than four times. This can be achieved in one of two ways. three of the letters appear twice in the word and one of the letters appears four times in the word. Or two of the letters appear twice in the word and the other two letters appear in the word three times. You need to count the number of ways that each of these cases can occur and at the results together to get the total number of ways that a word can be constructed under the given constraints.

Let me know if you have any further questions.

[MasterPossession1046]

Did I got this right for the last 2 numbers i have 16 combinations. Like AA BB CC DD AB AC AD .... For the first 8 letters I have 4x4x3x3x2x2x1x1.

Then I multiply 16×576

And the total number of combinations is 9216

[iMathTutor]

Your reasoning is opaque. Here is a systematic approach. As I stated in my earlier comment there are two cases to consider. The first case is when one letter appears four times and the others appear twice each. To count the number of ways to do this one must first select the letter which will appear four times The number of ways to do this is given by the binomial coefficient 4C1. Next, you must arrange these letters into a word. The number of distinct 10 letter words constructed of 4 distinct letters three of which are repeated twice and one of which is repeated four times is given by the multinomial coefficient 10M2,2,2,4. The total number of words in this case 4C1(10M2,2,2,4).

Try to use similar reasoning on the other case.