Misc 6
The English alphabet has 5 vowels and 21 consonants. How many words with two different vowels and 2 different consonants can be formed from the alphabet?
Number ways of selecting 2 vowels & 2consonants
= 5C2 × 21C2
= 5!/(2!(5 − 2)!) × 21!/(2!(21 − 2)!)
= 5!/2!3! × 21!/2!19! = (5 × 4 × 3!)/(2 × 1 × 3!) × (21 × 20 × 19!)/(2 × 1 × 19!) = 10 × 210
= 2100
Hence,
Total number of ways of selecting 2 vowels and 2 consonants = 2100
We have selected the letters,
Now, we have to arrange
Number of arrangements of 4 letters
Number of arrangements of 4 letters = 4P4
= 4!/(4 − 4)!
= 4!/0! = 4!/1
= 4 × 3 × 2 × 1 = 24 ways
Thus,
Total number of words
= Number of ways of selecting × Number of arrangements
= 2100 × 24
= 50400

Made by

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.