Problem: How many different arrangements using 5 letters of the word INTEGRAL, if no two vowels are adjacent?
I approached this question by separating it into scenarios: -no vowels -one vowel -two vowels but not adjacent -three vowels but not adjacent I only had trouble trying to calculate the scenario of 2 and 3 vowels, as my answer was incorrect. Can anyone help out?
There are 3 vowels: i,e,a
and 5 consonants: n,t,g,r,l.
For making 5 letter words, you have following choices: 3 vowels, 2 vowels, 1 vowel, 0 vowel.
For 3 vowels: you have 3 vowels, select 2 consonants from 5 in C(5,2) ways.
Arrange in VCVCV, wherein, vowels can inter arrange in 3! and consonants in 2! ways.
So, C(5,2)*3!*2!
For 2 vowels: select 2 vowels out of 3 in C(3,2) and 3 consonants from 5 in C(5,3) ways.
For arrangement, subtract those cases where vowels are consecutive(put 2 vowels in a box. now you have VV,C,C,C to be arranged in 4!*2!, 2! for vowel inter arrange) from total ways(5!); i.e. 5!-4!*2!.
Total is C(5,3)C(3,2)(5!-4!*2!).
For 1 vowel: select 1 vowel from 3 in C(3,1) and 4 consonants from 5 in C(5,4) ways.
Arrange in 5! ways.
Total is C(5,4)*C(3,1)*5!.
For 0 vowels: just arrange 5 consonants in 5! ways.
So,
total ways= C(5,2)*3!*2! + C(5,3)C(3,2)(5!-4!*2!) + C(5,4)*C(3,1)*5! + 5!