Question

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

Answer 1

Hint: In this question we need to find the number of distinct permutations of the word MISSISSIPPI where four I’s not come together. In order to do this, we will find the total permutation and subtract it from total permutation of I coming together. This will help us simplify the question and reach the answer.

Complete step-by-step answer:

We have to find the number of distinct permutations of the word MISSISSIPPI where four I's do not come together.

So, Total number of permutation of 4I not coming together = Total permutation – Total permutation of I’s coming together.

For total permutations in MISSISSIPPI,
As there are repeating characters in the word, so we will use the formula, $\dfrac{{n!}}{{p!q!r!...}}$

We are having 4I’s, 4S’s, 2P’s and 1M

So, the total number of permutations $= \dfrac{{11!}}{{4!4!2!}} = 34560$.

Now, for total permutations of I’s coming together in MISSISSIPPI,

We will take 4I’s as 1I,

So, the total permutations of I’s coming together $= \dfrac{{8!}}{{4!2!}} = 840$

As, the total number of permutation of 4I not coming together = Total permutation – Total permutation of I’s coming together

Therefore, the total permutation of 4I’s not coming together = 34650 – 840 = 33810.

Note: Whenever we face such types of problems the value point to remember is that we need to have a good grasp over permutations and its formulas. The basic formula to calculate permutations has been discussed above and used to solve the given question. We must also remember that the approach used above is the best way to solve these types of questions.