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

Solution:

We know that the number of arrangements (permutations) that can be made out of n things out of which there are p, q, r, ... number of repetitions = n! / [p! q! r! ...].

In the given word MISSISSIPPI,

No. of M's = 1
No. of I's = 4
No. of S's = 4
No. of P's = 2
Total number of letters = 11. 

This, the total number of arrangements possible with the given alphabets = 11! / (4! 4! 2!) = 34,650.

If we consider all the four I's one unit, then we get

M, S, S, S, S, P, P, \(\fbox{IIII}\).

Then total number of units is 8 among which S is repeated 4 times and P is repeated twice.

Then the total number of arrangements (permutations) possible with the given alphabets = 8! / (4! 2!) = 840.

The distinct number of permutations of the letters in MISSISSIPPI that do not have the four I’s together = 34,650 - 840 = 33,810

NCERT Solutions Class 11 Maths Chapter 7 Exercise 7.3 Question 10

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

Summary:

The distinct number of permutations of the letters in MISSISSIPPI that do not have the four I’s together 33,810