I'm not sure to how to correctly solve it, because I've never learned combinatorics.
At first I thought how many combinations of 3 letter word are possible, so I think it is $\binom{6}{3}=120$, then in each word I get I can change the order of the letters so it will be $3!$.
So is the answer $\binom{6}{3}\cdot3!=720$? Is the way ok?
Then if I want to add the words that can have the same letters 2-3 times, what then I do?
[Math] ABCDEF – All Combinations of 3 letter word
combinationscombinatorics
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Best Answer
If letters cannot repeat in a three-letter word, there are $6$ choices (A, B, C, D, E, or F) for the first letter. There are then $5$ choices for the second letter (the five letters that were not chosen in the first letter) and then there are $4$ choices for the third letter. This gives $6 \cdot 5 \cdot 4 = 120$ three-letter words with repeats not allowed.
If letters can be repeated as many times as you want, there are $6$ options (A, B, C, D, E, or F) for the first letter, second letter, and third letter. Then $6^3 = 216$ are the number of options for all three-letter-words.
If a letter can be repeated at most twice, it gets more complicated. Notice that a three-letter word has all different letters, two letters that are the same and one that is different, or all three letters the same. Without any restrictions on the number of repetitions, we found $216$ three-letter words. Of those, there are exactly $6$ which have letters repeated $3$ times (AAA, BBB, CCC, DDD, EEE, and FFF). That means the other $216 - 6 = 210$ three-letter-words have letters repeated at most twice.