Find the number of combinations of n letters out of 3n letters of which n are A and n are B and rest are unlike

Question

I think I should be going by partition method. Am I approaching in the right direction? Please give an idea. And please don't solve it i want to solve.I am further claiming that the equation has something to do with non negative integral solution. Please suggest some idea.

Answer 1

You can decide to choose $r\in[0..n]$ of the different letters.The remaining $n-r$ places can be filled with As and Bs in $n-r+1$ different ways. It follows that there are $$\sum_{r=0}^n{n\choose r}(n-r+1)=(n+2)\,2^{n-1}$$ different combinations (Mathematica computed the sum for me).