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Question
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?
Solution
2 different vowels and 2 different consonants are to be selected from the English alphabet.
Since there are 5 vowels in the English alphabet, number of ways of selecting 2 different vowels from the alphabet
= `""^5C_2 = (5!)/(2!3!) = 10`
Since there are 21 consonants in the English alphabet, number of ways of selecting 2 different consonants from the alphabet
= `""^21C_2 = (21!)/(2!19!) = 210`
Therefore, number of combinations of 2 different vowels and 2 different consonants = 10 × 210 = 2100
Each of these 2100 combinations has 4 letters, which can be arranged among themselves in 4! ways.
Therefore, required number of words = 2100 × 4!
= 24 x 2100
= 50400
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