Question

How many words each of 3 vowels and 2 consonants can be formed from the letters of the word INVOLUTE?

Answer 1

Given as

The word ‘INVOLUTE’

The total number of letters = 8

The total vowels are = I, O, U, E

The total consonants = N, V, L, T

Therefore number of ways to select 3 vowels is ⁴C₃

And the number of ways to select 2 consonants is ⁴C₂

Then, the number of ways to arrange these 5 letters = ⁴C₃ × ⁴C₂ × 5!

By using the formula,

ⁿC_r = n!/r!(n – r)!

⁴C₃ = 4!/3!(4 - 3)!

= 4!/(3! 1!)

= [4 × 3!] / 3!

= 4

⁴C₂ = 4!/2!(4 - 2)!

= 4!/(2! 2!)

= [4 × 3 × 2!] / (2! 2!)

= [4 × 3] / (2 × 1)

= 2 × 3

= 6

Therefore, by substituting the values we get

⁴C₃ × ⁴C₂ × 5! = 4 × 6 × 5!

= 4 × 6 × (5 × 4 × 3 × 2 × 1)

= 2880

∴ The no. of words that can be formed containing 3 vowels and 2 consonants chosen from ‘INVOLUTE’ is 2880.