Multiple choice

How many different words can be formed from $12$ consonants and $5$ vowels by taking $4$ consonants and $3$ vowels in each word?

  1. $4950\times7!$
  2. $7!^2$
  3. $^{12}C_4\times ^5C_3$

  4. none of these
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A Correct answer
Explanation

Ways to choose 4 consonants from 12 is 12C4. Ways to choose 3 vowels from 5 is 5C3. Total ways to form words = (12C4 * 5C3) * 7!. 12C4 = 495, 5C3 = 10. 4950 * 7!.

AI explanation

Using the combination formula, select 4 consonants from 12 in 12C4 ways and 3 vowels from 5 in 5C3 ways. Since each formed word contains exactly 7 letters, they can be permuted among themselves in 7! ways. The total number of words is the product 12C4 * 5C3 * 7!, where 12C4 is 495 and 5C3 is 10, yielding 4950 * 7!.

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