Q167 / 269 · NDA 2025 (Phase 1) · Mathematics
In how many ways can the letters of the word DELHI be arranged keeping the positions of vowels and consonants unchanged?
- A$6$
- B$9$
- Correct answerC$12$
- D$24$
Solution
The word DELHI consists of $5$ letters: $3$ consonants (D, L, H) and $2$ vowels (E, I).
The arrangement of letters in the word is Consonant-Vowel-Consonant-Consonant-Vowel.
Since the positions of vowels and consonants must remain unchanged, the consonants can only be placed in the $1$st, $3$rd, and $4$th positions, and the vowels can only be placed in the $2$nd and $5$th positions.
The $3$ consonants can be arranged in their $3$ positions in $3!$ ways.
The $2$ vowels can be arranged in their $2$ positions in $2!$ ways.
Total number of arrangements = $3! \times 2! = 6 \times 2 = 12$.
Answer: $12$
The arrangement of letters in the word is Consonant-Vowel-Consonant-Consonant-Vowel.
Since the positions of vowels and consonants must remain unchanged, the consonants can only be placed in the $1$st, $3$rd, and $4$th positions, and the vowels can only be placed in the $2$nd and $5$th positions.
The $3$ consonants can be arranged in their $3$ positions in $3!$ ways.
The $2$ vowels can be arranged in their $2$ positions in $2!$ ways.
Total number of arrangements = $3! \times 2! = 6 \times 2 = 12$.
Answer: $12$
Hence, the correct answer is option: 'C'.