Multiple choice

Let E denote the set of letters of the English alphabet, V = {a, e, i, o, u}, and C be the complement of V in E. Then the number of four-letter words (where repetitions of letters are allowed) having at least one letter from V and at least one letter from C is

  1. 2,61,870

  2. 3,14,160

  3. 4,25,880

  4. 8,51,760
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A Correct answer
Explanation

Total 4-letter words with 26 letters = 26^4 = 456,976. Words with no vowels (all consonants) = 21^4 = 194,481. Words with no consonants (all vowels) = 5^4 = 625. Words with at least one vowel and at least one consonant = Total - (All Vowels) - (All Consonants) = 456,976 - 625 - 194,481 = 261,870.

AI explanation

Let E be the 26-letter alphabet, V be the set of 5 vowels, and C be the set of 21 consonants. The total number of four-letter words with repetitions allowed is 26 raised to the power of 4, which is 456976. We subtract the words made only of consonants, 21 raised to the power of 4 or 194481, and the words made only of vowels, 5 raised to the power of 4 or 625. The calculation is 456976 minus 195106, resulting in 261870.

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