Multiple choice

The number of words of four letters containing equal number of vowels and consonants, where repetition is allowed, is

  1. $105^2$
  2. $210\times 243$
  3. $150\times 243$
  4. $150\times 21^2$
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D Correct answer
Explanation

For a 4-letter word with equal vowels (V) and consonants (C), we need 2 V and 2 C. There are 5 vowels and 21 consonants. The number of ways to pick positions for 2 vowels is 4C2 = 6. The number of ways to fill these positions is (5^2) * (21^2). Total = 6 * 25 * 441 = 150 * 441 = 150 * 21^2.

AI explanation

A four letter word with equal vowels and consonants must have exactly 2 vowels and 2 consonants. The vowels can be chosen in 5C2 ways and the consonants in 21C2 ways, which is 10 times 210 equaling 2100 combinations. Since repetition is allowed, each of the 4 chosen positions can be any of its 3 available letters, giving 3^4 arrangements per combination. The total number of words is 2100 times 81, which simplifies to 150 times 21^2.

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