COMEDK 2025 Evening Solved Paper

Concept:Treat the word as a combination of consonants and vowel blocks to satisfy the given condition.Explanation:The word "DEFINITE" has 8 letters: D, E, F, I, N, I, T, E.Consonants: D, F, N, T (4 distinct).Vowels: E, E, I, I (two identical E's and two identical I's).First, arrange the 4 consonants in a row. Number of ways = $4! = 24$.There are 5 gaps around the consonants: before the first, between each pair, and after the last.We must place two vowel-blocks (each block contains exactly two vowels) in two different gaps so that the blocks are separated.Number of ways to choose 2 gaps out of 5 = $\binom{5}{2} = 10$.Now arrange the 4 vowels into two blocks of two letters each. Since the vowels are not all distinct, we count the distinct sequences of 4 vowels (E,E,I,I) in the two blocks. Number of such sequences = $\frac{4!}{2!\,2!} = 6$.Total number of words = (consonant arrangements) × (gap choices) × (vowel arrangements) $= 24 \times 10 \times 6 = 1440$.Answer:1440 (Option C)