How many possibilities are there to arrange the letters of the word MISSISSIPPI, such that no S is next to another S?
Here are some possible examples:
Note that there is never something like SSS.
Or in other words: No S touches another S!
The easy part is to figure out the possibilities for the letters M, I and P:
Those three letters occupy 7/11 places in the whole word. So we have 7 places to put M, then there are 6 places left to place 4xI and lastly there are 2 places left to place 2xP: $\binom{7}{1}*\binom{6}{4}*\binom{2}{2}$
But we still have to place four times S.
Currently our word has 7 letters (M,I,P) and thus looks like $\text{XXXXXXX}$ (where $X\in\{M,I,P\}$). And we have 8 places to put the four S into: $\text{sXsXsXsXsXsXsXs}$. Which means we have $\binom{8}{4}$ possibilities to place the four S.
$$\binom{7}{1}*\binom{6}{4}*\binom{2}{2}*\binom{8}{4}=7350$$
MISSISSIPPI,arrange all other letters other than S. MIIIIPP can be done in $$\frac{7!}{(4!*2!)}$$ and then insert those $4$ S’s in the 8 gaps of the above formed word, that means we are choosing $4$ gaps out of $8$ in $8C4$ ways so total is $8C4\times105=70\times105=7350$
