The number of 4-letter words, with or without meaning, each consisting of 2 vowels and 2 consonants, which can be formed from the letters of the word UNIVERSE without repetition is __________.

Solution

Maths sir · 4 steps

Pick the distinct letters first, then arrange them: choose, choose, and multiply by 4!.

1. Sort the letters. UNIVERSE has vowels U, I, E, E and consonants N, V, R, S. Since no letter may repeat in the word, the two E's give only one usable E, so there are 3 distinct vowels (U, I, E) and 4 distinct consonants.
2. Choose the letters. Choose 2 of the 3 vowels in C(3, 2) = 3 ways and 2 of the 4 consonants in C(4, 2) = 6 ways. Each choice gives 4 different letters.
3. Arrange them. Four different letters can be arranged in 4! = 24 ways.

   C(3, 2) × C(4, 2) × 4! = 3 × 6 × 24 = 432

4. So. The number of 4-letter words is 432.

The common slip: 504. Also counting words that use E twice (the pair EE with 2 consonants adds 6 × 4!/2! = 72) gives 504, but 'without repetition' rules that out.

The idea to keep. For word problems, first select the letters with combinations, then arrange them with factorials. Before you count, list the distinct letters, because repeated letters shrink your choices.

Answer from the NTA final answer key. Solution by PrepWiser.