Combinatorics of given alphabet I'm looking for the formula to determine the number of possible words that can be formed with a fixed set of letters and some repeated letters. For instance take the 8-letter word SEASIDES and find out how many possible anagrams can be made: there are 3x S, 2x E, and 1 each of A, I and D; words of less than 8 letters are allowed.
The formula for all 8 letters is not too difficult W8 = 8! / (3! * 2!). However, this doesn't include shorter anagrams, which seem a less intuitive formula.

You can use $$\displaystyle \sum_{s=0}^3 \sum_{e=0}^2 \sum_{a=0}^1  \sum_{i=0}^1 \sum_{d=0}^1 \dfrac{(s+e+a+i+d)!}{s!\,e!\,a!\,i!\,d!}$$ and if you like ignore the $ a!\,i!\,d!$ which is always $1$. This will give $9859$ possibilities. You may want to subtract $1$ if you want to exclude $0$-letter anagrams, leaving $9858$ possibilities.
For the full $8$-letter anagrams, each index takes its maximum value, giving $\frac{8!}{3! \, 2!}=3360$ possibilities.  It seems there are another $3360$ $7$-letter anagrams (is this a co-incidence?), so between them more thatn two-thirds of the total.
