Question

Find the number of ways in which the letters of the word $MISSISSIPPI$ can be arranged if atleast one vowel is separated from rest of the vowels

Answer 1

These are 11 digit in MISSISSIPPI

Total permutations $\frac{11!}{4!4!2!}$

when all vowels are together IIII

$=\frac{8!}{4!2!}$

$\therefore$ No of ways in which the letters can be arranged if atleast one vowel is separated from rest of the vowel

$=\frac{11!}{4!4!2!}-\frac{8!}{4!2!}$

$=\frac{11\times 10\times 9\times 8!}{4\times 3\times 2\times 1\times 4!2!}-\frac{8!}{412!}$

$=\frac{8!}{4!2!}[\frac{11\times 10\times 9}{4\times 3\times 2}-1]$

$=\frac{8!}{4!2!}[\frac{11\times 15}{4}-1]$

$=\frac{8!}{4!2!}\left[\frac{165-4}{4}\right]=\frac{8!\times 161}{4!2!\times 4}$

$=\frac{8\times 7\times 6\times 5\times 4!\times 161}{4!2!\times 4}$

= 33,810 ways.