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# How many different words, each containing 2 vowels and 3 consonants can be formed with 5 vowels and 17 consonants?

Given that we need to find the no. of words formed by 2 vowels and 3 consonants which were taken from 5 vowels and 17 consonants.

Let us find the no. of ways of choosing 2 vowels and 3 consonants and assume it to be N1.

N1 = (No. of ways of choosing 2 vowels from 5 vowels) × (No. of ways of choosing 3 consonants from 17 consonants)

N1 = (5C2) × (17C3)

We know that ,

And also n! = (n)(n – 1)......2.1

N1 = 10 × 680

N1 = 6800

Now we need to find the no. of words that can be formed by 2 vowels and 3 consonants.

Now we need to arrange the chosen 5 letters. Since every letter differs from other.

The arrangement is similar to that of arranging n people in n places which are n! ways to arrange. So, the total no. of words that can be formed is 5!.

Let us the total no. of words formed be N.

N = N1 × 5!

N = 6800 × 120

N = 816000

The no. of words that can be formed containing 2 vowels and 3 consonants are 816000.

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