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Out of 7 consonants and 4 vowels, how many words can be formed such that it contains 3 consonants and 2 vowels ?
1. 36000
2. 55000
3. 25200
4. 75000
5. None of these

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Correct Answer - Option 3 : 25200

Concept:

The number of ways to select r things out of n given things wherein r ≤ n is given by: \({\;^n}{C_r} = \frac{{n!}}{{r!\; × \left( {n - r} \right)!}}\)

Calculation:

Given: There are 7 consonants and 4 vowels

Here, we have to find how many word can be formed such that it has 3 consonants and 2 vowels.

No. of ways to select 2 vowels out of 4 vowels = \({\;^4}{C_2} \)

No. of ways to select 3 consonants out of 7 consonants = \({\;^7}{C_3} \)

∴ No. of words that can be formed which contains 3 consonants and 2 vowels = \({\;^4}{C_2} \) × \({\;^7}{C_3} \)

As we know that, \({\;^n}{C_r} = \frac{{n!}}{{r!\; × \left( {n - r} \right)!}}\)

⇒  \({\;^4}{C_2} \) × \({\;^7}{C_3} \)  = 6 × 35 = 210

The no. of ways to arrange words containing 3 consonants and 2 vowels = 210 × 5! = 25200

Hence, option C is the correct answer.

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