Question

In how many of the distinct permutations of the letters in POSSIBILITY do the 3 I's not come together?​

Answer 1

Given:

The letter POSSIBILITY.

To Find:

In how many ways to arrange the letters of  POSSIBILITY so that 3 I's does not come together?​

Solution:

Here, in word POSSIBILITY

there are three I's, so, taking all the I's as one letter we have, 8 letter word to arrange.

So we can arrange 8 letter word in 8! ways , but we already have three I's and two S's , so, we have to divide it by 3!2!

⇒ letter of ‘POSSIBILITY’ be arrange so that all I's are come together

  $=\dfrac{8!}{3!2!} \\=3360$

Total number of letters in word POSSIBILITY = 11

so, number of ways to arrange the letters of the word POSSIBILITY = 11 !

but we already have three I's and two S's , so, we have to divide it by 3!2!

Thus, number of ways to arrange the letters of the word POSSIBILITY                   $=\dfrac{11!}{3!2!}\\=3326400$

Hence,

Total number of ways the letters of  POSSIBILITY is arranged so that 3 I's does not come together = number of ways to arrange the letters of the word POSSIBILITY - number of ways the letter of ‘POSSIBILITY’ be arrange so that all I's are come together

$= 3326400-3360$

$=3323040$ .