How many words with two different vowels and 2 different consonants can be formed from the alphabet ?

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 ?

Answer 1

We have 5 vowels and 21 consonants.

select : 2 vowels and 2 consonants

We can choose 2 vowels from 5 in ⁵C₂ ways = 10 ways

We can choose 2 consonants from 21 in ²¹C₂ ways = 210 ways

Now these selected 4 letters can be arranged in 4! ways = 24 ways

By fundamental theorem,

total number of ways(words) = 10*210*24

                                       = 50400