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

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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