Given:

The word ‘INVOLUTE’

Total number of letters = 8

Total vowels are = I, O, U, E

Total consonants = N, V, L, T

So number of ways to select 3 vowels is ^{4}C_{3}

And numbre of ways to select 2 consonants is ^{4}C_{2}

Then, number of ways to arrange these 5 letters = ^{4}C_{3} × ^{4}C_{2} × 5!

By using the formula,

^{n}C_{r} =\(\dfrac{ n!}{r!(n – r)!}\)

^{4}C_{3} = \( \dfrac{4!}{3!(4-3)!}\)

=\(\dfrac{ [4×3!] }{ 3!}\)

= 4

^{4}C_{2} = \(\dfrac{ 4!}{2!(4-2)!}\)

= \(\dfrac{ (4×3×2!) }{(2! 2!)}\)

= 2 × 3

= 6

So, by substituting the values we get

^{4}C_{3} × ^{4}C_{2} × 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.

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