# Thread: (Simple?) combination problem (proof required)

1. ## (Simple?) combination problem (proof required)

How many different combinations of 4 letters can be made out of the letters in the word MATHEMATICS?

I took a guess at it myself, calculation being 11^4, no idea if it's correct though. How would I prove this if it is?

2. Originally Posted by JacobSkylar
How many different combinations of 4 letters can be made out of the letters in the word MATHEMATICS?
The difficulty here is the repeated letters: $\displaystyle M_2A_2T_2HEICS$.
We could have “TCIS”, “ATTS”, or even “MMAA”.
So we need to count three possibilities.
1) All four letters are different: $\displaystyle ^8\mathcal{P}_4$ (permutation of 8 letters, 4 at a time)
2) Exactly two repeated letters. Like "ATTS"
3) Two letters each repeated. Like "MMAA"

No you try to finish.

3. Originally Posted by Plato
The difficulty here is the repeated letters: $\displaystyle M_2A_2T_2HEICS$.
We could have “TCIS”, “ATTS”, or even “MMAA”.
So we need to count three possibilities.
1) All four letters are different: $\displaystyle ^8\mathcal{P}_4$ (permutation of 8 letters, 4 at a time)
2) Exactly two repeated letters. Like "ATTS"
3) Two letters each repeated. Like "MMAA"

No you try to finish.
I made a small mistake in the explanation of the task. It is supposed to be different arrangements of letters, not combinations, meaning it could be like MMAA and AAMM. The order of the letters just has to be different.

However, I see where you're coming at, but i'm not sure if I can figure this out on my own still. I'm the type of person that learns how to do calculations by looking at the end result and making judgments based on that, to use in the future.

Thanks a bunch for your effort thus far!

4. Originally Posted by Plato
The difficulty here is the repeated letters: $\displaystyle M_2A_2T_2HEICS$.
We could have “TCIS”, “ATTS”, or even “MMAA”.
So we need to count three possibilities.
1) All four letters are different: $\displaystyle ^8\mathcal{P}_4$ (permutation of 8 letters, 4 at a time)
2) Exactly two repeated letters. Like "ATTS"
3) Two letters each repeated. Like "MMAA”
Originally Posted by JacobSkylar
I made a small mistake in the explanation of the task. It is supposed to be different arrangements of letters, not combinations, meaning it could be like MMAA and AAMM. The order of the letters just has to be different.
I beg you pardon. I did not misunderstand your question.
But you totally did not understand my reply.

Using the letters $\displaystyle m,m,a,a$ there are $\displaystyle \frac{4!}{(2!)(2!)}$ ways to rearrange that string.
But there are $\displaystyle \binom{3}{2}=3$ ways to have such a string.

I am not sure that you understand this question. Do you?

5. Ok, here we go:

1/ There can be only 1 of each letter = 8! = 40320 permutations.

2/ There are two M's but 1 only of every other = 4 on top of 2 * 7 * 6 = 252.

3/ There are two A's but 1 only of every other = 4 on top of 2 * 7 * 6 = 252.

4/ There are two T's but 1 only of every other = 4 on top of 2 * 7 * 6 = 252.

5/ There are two M's and two A's = 4 on top of 2 * 1 = 6

6/ There are two M's and two T's = 4 on top of 2 * 1 = 6

7/ There are two T's and two A's = 4 on top of 2 * 1 = 6

Totalling in 41094. Am I way off or am I on the right track?

6. Originally Posted by JacobSkylar
Totalling in 41094. Am I way off or am I on the right track?
Correct! Way to go.