Let's say there are 7 available letters: ABCDEFG, and I would like to calculate the number of unique 5 letter combinations that can be made, e.g. AABBE. Double letters are perfectly fine, the only thing is that the order doesn't matter, so AABBA and ABAAB are regarded as the same and are counted only once.
I think the binomial theorem can be used when none of the letters are to be repeated, but I'm a bit stuck on adding the permutations where a letters is used more than once.
Cheers!
Let $x_{1},x_{2},...x_{7}$ denote the number of letter $A,B,C,...,G$ respectively. We need to find non - negative integer solution of $\sum_{i=1}^{7}{x_{i}}=5$.
This is stars and bars problem with solution $\binom{11}{6}$