Question

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

Answer 1

Answer

The word MISSISSIPPI have 11 letters,

Here,

M ➝ 1 time

I ➝ 4 times

S ➝ 4 times

P ➝ 2 times

The number of permutations of the word MISSISSIPPI in which 4I's and 4S's are alike so,

$\sf \dfrac{11!}{4!4!2!} \: \: \: ...(1)$

If all the I's are together, then it will be considered as one letter and remaining 7 letters and 1I's letter will be considered as 8 letter.

So, the number of permutations is $\sf \dfrac{8!}{4!2!}$

Thus, the total number of arrangements

$\sf = \dfrac{11!}{4!4!2!} - \sf \dfrac{8!}{4!2!}$

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

$\sf = 34650 - 840$

$\sf = 33810$