Question

In how many of the distinct permutations of the letters in MISSISSIPPI do the four 'I's not come together?

Answer 1

Total letters of the word MISSISSIPPI = 11.

Here $M=1,I=4,S=4$ and $P=2$

$\therefore$ Number of permutations = $\frac{11!}{4!4!2!}$

= $\frac{11\times 10\times 9\times 8\times 7\times 6\times 5\times 4!}{4!\times 4\times 3\times 2\times 1\times 2\times 1}$

= $34650$

When the four 'I's come together, then it becomes one letter so total number of letters in the word when all I"s come together = 8.

$\therefore$ Number of permutations

= $\frac{8!}{4!2!}=\frac{8\times 7\times 6\times 5\times 4!}{4!\times 2\times 1}=840$

Number of permutations when four I"s do not come together

= $34650-840=33810.$