Q:

Selection Problem

How many ways are there to select 8 letters from 24 letters where 8 letters are 'a' ,8 letters are 'b'and rest are all unlike?

A:

Break into cases based on how many unlike letters were selected.
Supposing that you selected $k$ of the unlike letters, pick which $k$ of those they were that were selected in $\binom{8}{k}$ ways.
Then, for the remaining $8-k$ letters that you were going to select, choose how many of those will be a's vs b's in $8-k+1$ ways.
$$\sum\limits_{k=0}^8\binom{8}{k}(8-k+1) = 9\times 2^8 - 8\times 2^7 = 1280$$

