The formula you are using is for combinations where order of the letters does not matter. Of course, for words order does matter, so we need to treat the groupings as permutations. (Note, we are treating a "word" as any group of letters whether it makes linguistic sense or not.)

To determine the number of possible words simply decide how many letters you have for the first letter, say 6, then take that letter out and determine how many for the second with what is left over, say 5, and then finally for the third letter you likely have 4 letters left over. So the total number of three letter words would be 6*5*4 = 120. For four letter words, 6*5*4*3 = 360, etc.

Note this does assume you have 6 different letters. If you have 5 distinct letters, say for the word "assume". The double S's change things.

For this, let's go back to the combinations you were using. If we choose 3 letters from the six without regard to order, we get 6!/{3!*3!} = 20 as you found before. That's 20 ways to pick 3 letters. If we then worry about order, there are 6 ways to order the three letters, 3*2*1 as we discussed above. Thus 20*6 = 120 gets us back to our previous answer of number of three letter words from 6 letters.

Let's use this info to our advantage when our 6 letters are "assume". 20 ways to pick three letters. How many of these ways have the two S's? Remember order doesn't matter at the moment, so there are only 4 ways we can pick three letters containing both S's. This is obvious since we have only four ways to pick the third letter.

Now consider these four special collections containing 2 S's and a third letter. How many distinct ways to arrange them? Hopefully, it is somewhat obvious that there are three ways. (Basically, choose where the odd letter is in the order and the other two spots have to be S's.) Four of these groups, each with three ways to arrange them, means we have 12 distinct words that can be arranged.

The remaining 16 groups each have 3 distinct letters and we already know that each of these groups makes 6 words. 16 * 6 = 96, add in the 12 from above and we get 108 distinct three letter words that can be made from 6 letters with one repeat.

Hopefully, by mimicking the above process you can figure out the number of four, five and six letter words, taking into account for the possible times when you have letters that repeat.

Good luck!

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