Question

A word has 8 consonants and 3 vowels. How many distinct words can be formed if 4 consonants and 2 vowels are chosen?

Answer 1

4 consonants can be selected from 8 consonants in ⁸C₄ ways and 2 vowels can be selected from 3 vowels in ³C₂ ways.

∴ the number of words with 4 consonants and 2 vowels = ⁸C₄ × ³C₂

= `(8!)/(4!4!) xx (3!)/(2!1!)`

= `(8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1) xx (3 xx 2!)/(2!)`

= 70 × 3

= 210

Now each of these words contains 6 letters which can be arranged in ⁶P₆ = 6! ways.

∴ the total number of words that can be formed with 4 consonants and 2 vowels

= 210 × 6!

= 210 × 6 × 5 × 4 × 3 × 2 × 1

= 151200.