Number of Four Letter Words Formed From Inconsequential

To find the number of 4-letter words consisting of two vowels and two consonants formed from the letters of the word INCONSEQUENTIAL without repeating any letter in the formed word, we first analyze the available letters in the given word.

Step 1: Categorize the letters of "INCONSEQUENTIAL"

Let's count the occurrences of each letter and divide them into vowels and consonants:

1. Vowels:

   • $\text{I}$ appears 2 times
   • $\text{O}$ appears 1 time
   • $\text{E}$ appears 2 times
   • $\text{U}$ appears 1 time
   • $\text{A}$ appears 1 time

   The set of distinct vowels available is $\{A, E, I, O, U\}$, which gives $5$ distinct vowels.

2. Consonants:

   • $\text{N}$ appears 3 times
   • $\text{C}$ appears 1 time
   • $\text{S}$ appears 1 time
   • $\text{Q}$ appears 1 time
   • $\text{T}$ appears 1 time
   • $\text{L}$ appears 1 time

   The set of distinct consonants available is $\{C, L, N, Q, S, T\}$, which gives $6$ distinct consonants.

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Step 2: Selection and Arrangement

Since no letter can be repeated in the 4-letter word:

1. Selecting 2 distinct vowels from the 5 available distinct vowels: $\binom{5}{2} = \frac{5 \times 4}{2} = 10$

2. Selecting 2 distinct consonants from the 6 available distinct consonants: $\binom{6}{2} = \frac{6 \times 5}{2} = 15$

3. Arranging the 4 selected distinct letters to form a 4-letter word: $4! = 24$

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Step 3: Total Number of Words

By the fundamental principle of counting, the total number of such 4-letter words is:

$\text{Total Words} = \binom{5}{2} \times \binom{6}{2} \times 4! = 10 \times 15 \times 24 = 3600$

Thus, the correct option is D (3600).