Two ways to find the number of arrangements of MISSISSIPPI such that $S$'s are not placed next to each other?

Question

In how many arrangements of MISSISSIPPI the $S$'s are not placed next to each other?

Assume we only have the word MIIIPPI,there are $8$ places for which a S can be placed in,on the other hand there are permutations of the first word,follows the number of such arrangements is :$$\binom{8}{4}\frac{7!}{2!\cdot4!}$$

I tried to use another method,the number of such arrangements is the total arrangements minus the number of the arrangements for which such $S$ are all consecutive,so : $$\frac{11!}{4!4!2!}-8\frac{7!}{4!2!}$$

But the answers are not the same,so where was I wrong?

Answer 1

The first part is fine. We have \begin{align*} \binom{8}{4}\frac{7!}{4!2!1!}=\color{blue}{7\,350}\tag{1} \end{align*} The number of all different words which can be built from the letters in $MISSISSIPPI$ is \begin{align*} \frac{11!}{4!4!2!1!}=34\,650\tag{2} \end{align*} From these words we have to subtract the words which contain consecutive $S$. We consider the $8$ positions where we can place the $4$ S.

\begin{align*} \,_{1}\,M\,_{2}\,I\,_{3}\,I\,_{4}\,I\,_{5}\,P\,_{6}\,P\,_{7}\,I\,_{8}: \end{align*}

We have the following ways to group $4$ S with the following number of placements per group \begin{align*} &(1,1,1,1)&&(S,S,S,S)\\ &(1,1,2)&&(S,S,SS)\ \ \to3\\ &(2,2)&&(SS,SS)\ \ \ \ \to1\\ &(1,3)&&(S,SSS)\ \ \ \ \to2\\ &(1)&&(SSSS)\ \ \ \ \ \ \to1 \end{align*} Since only the first group $(1,1,1,1)$ representing $4$ S-runs of length $1$ is admissible, we have to subtract from (2) the number of words built from the other groups.

• The size of the group: $|(S,S,SS)|=3$ for instance. So, we have $\binom{8}{3}$ ways to place the members of the group and $3$ different ways to arrange the three members within a placement.

We obtain \begin{align*} &\frac{11!}{4!4!2!1!}-\frac{7!}{4!2!1!}\left(3\binom{8}{3} +\binom{8}{2}+2\binom{8}{2}+\binom{8}{1}\right)\\ &\qquad=34\,650-105\left(56\cdot3+28\cdot1+28\cdot2+8\cdot 1\right)\\ &\qquad=34\,650 -26\,670\\ &\qquad\,\,\color{blue}{=7\,350} \end{align*} in accordance with (1).