The English alphabet has 5 vowels and 21 consonants. How many words with 2 different vowels and 2 different consonants can be formed from the alphabet?

Explanation

Total number of ways of choosing
(2 consonants out of 21) and (2 vowels out of 5)
`=.^(21)C_(2)xx.^(5)C_(2)=((21xx20)/(2xx1)xx(5xx4)/(2xx1))=2100`.
Number of groups, each containing 2 consonants and 2 vowels
=2100.
Each group contains 4 letters.
Number of ways of arranging 4 letters amongst themselves
`=4!=(4xx3xx2xx1)=24`.
Hence, the required number of words `=(2100xx24)=50400`.