Question

The number of 4-letter words, with or without meaning, each consisting of two vowels and two consonants that can be formed from the letters of the word INCONSEQUENTIAL, without repeating any letter, is:

• A. 2670
• B. 2840
• C. 2920
• D. 3600

Hint:

First, find the set of distinct vowels and distinct consonants in the word. Then, use combinations to choose the required number of each and multiply by 4! to arrange them into words.

Answer 1

The Correct Option is D

The goal is to select 2 vowels and 2 consonants from the unique letters of 'INCONSEQUENTIAL' and arrange them into 4-letter words.

Vowels present: I, O, E, U, A. (Count: 5 unique vowels)
Consonants present: N, C, S, Q, T, L. (Count: 6 unique consonants)

Number of ways to select 2 vowels from 5: $^5C_2 = 10$.
Number of ways to select 2 consonants from 6: $^6C_2 = 15$.

Once we have chosen our 4 letters, they are all distinct (since the problem states 'without repeating any letter'). The number of ways to arrange 4 distinct items in a row is given by $P(4,4) = 4!$.

Total arrangements = (Selection of Vowels) $\times$ (Selection of Consonants) $\times$ (Arrangement of 4 letters)
Total arrangements = $^5C_2 \times ^6C_2 \times 4!$
Total arrangements = $10 \times 15 \times 24 = 3600$.