CONTEXT: Question made up by uni lecturer.
How many distinct arrangements of the letters in the word 'Mississippi' have no S's in the first six places?
My first attempt of the question involved treating the S's as a single block of letters (i.e. you find arrangements of elements of $\{M, I, I, I, I, P, P, SSSS\}$), however, this doesn't work because if the first six places can't contain S's, then the last five places will, which means the S's could be separated (i.e. in 'PIMIPISSISS', the block of S's is separated by a 'I').
My second attempt involved subtracting the complement (all distinct arrangements with S's only in the first six places) from the total number of distinct arrangements which is $\frac{11!}{4!4!2!}=34650$, but I realised this would require a lot more calculations than just considering the original case.
I'm a bit stuck on how to approach this question; any guidance would be greatly appreciated.
Best Answer
The word MISSISSIPPI has eleven letters. Therefore, if no S's appear in the first six places, they must occupy four of the last five positions. Choose which four of those five positions they occupy. That leaves seven positions to fill with four I's, two P's, and one M. Choose four of those seven positions for the I's, two of the remaining three positions for the P's, then fill the final position with the M.