Question

What is the probability that four S's come consecutively if all the letters of the word 'MISSISSIPPI' are rearranged randomly?

Answer 1

$MISSISSIPPI\Rightarrow 11$ letters $(4-S,4-I,2-P,1-M)$

Total no of ways of arranging $MISSISSIPPI\Rightarrow \frac{11!}{(4!)(4!)(2!)}=\frac{11!}{(4!)(4!)(2!}$

now, we need all $s$ to be together

Put all $s$ together and consider it as a block

$SSSS$

Then no. of letters will be $8(4-I,2-P,1-M,1block)$

no. of ways of such arrangement

$\Rightarrow \frac{8!}{(4!)(2!)}\Rightarrow \frac{8!}{2(4!)}$

Probability that all $4^{\prime }S$ are together $=\frac{\left(2\right)}{\left(1\right)}$

$=\frac{8!}{2\times (4!)}\times \frac{(4!)(4!)(2!)}{11!}$

$=\frac{8!}{11!}\times 4!$

$=\frac{(4\times 3\times 2\times 1)\times 8!}{11\times 10\times 9\times 8!}$

Probability that all ${}^{\prime }{s}^{\prime }$ are together $=\frac{4}{165}$