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?
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Are we selecting any random letter – Aderinsola Joshua Apr 05 '20 at 19:52
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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.
$$\sum\limits_{k=0}^8\binom{8}{k}(8-k+1) = 9\times 2^8 - 8\times 2^7 = 1280$$
JMoravitz
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@user768300 you could have picked $0$
a's, $1$a's, $2$a's, on up to $8-k$a's. Count those... maybe start counting from $1$ and then go back once you are done and then include $0$ as well. – JMoravitz Apr 05 '20 at 14:52 -