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

To find the total number of 4-letter words, we first identify the unique vowels and consonants available in the word 'INCONSEQUENTIAL'.

1. List the letters and counts:
The letters are: I(2), N(3), C(1), O(1), S(1), E(2), Q(1), U(1), T(1), A(1), L(1).

2. Categorize the distinct letters:
Since we are forming words 'without repeating any letter', we only care about the distinct letters in each category.
Distinct Vowels: {I, O, E, U, A}. There are 5 distinct vowels.
Distinct Consonants: {N, C, S, Q, T, L}. There are 6 distinct consonants.

3. Step-by-step selection:
- We need to choose exactly 2 vowels from the 5 distinct vowels available. The number of ways to do this is $^5C_2 = \frac{5 \times 4}{2 \times 1} = 10$ ways.
- We need to choose exactly 2 consonants from the 6 distinct consonants available. The number of ways to do this is $^6C_2 = \frac{6 \times 5}{2 \times 1} = 15$ ways.

4. Calculate total combinations:
The total number of sets of 4 letters (2 vowels and 2 consonants) we can form is $10 \times 15 = 150$.

5. Arrange the chosen letters:
Each set of 4 unique letters can be arranged in $4!$ (4 factorial) ways to form different words.
$4! = 4 \times 3 \times 2 \times 1 = 24$.

6. Final count:
Total words = (Number of selections) $\times$ (Number of arrangements)
Total words = $150 \times 24 = 3600$.