Question

The number of ways in which the letter of the word 'VERTICAL' can be arranged without changing the order of the vowels is 

• A. 6!×3!
• B. \(\frac{8!}{3}\)
• C. 6!×3
• D. \(\frac{8!}{3!}\)

Answer 1

The Correct Option is D

1. The word VERTICAL has 8 letters. 
2. It contains 3 vowels: E, I, and A.
3. We want to arrange the letters without changing the order of the vowels, so we focus on placing the consonants: V, R, T, C, and L.

We must choose 3 positions out of 8 to place the vowels (in fixed order), and fill the rest with the 5 consonants.

Number of ways to choose 3 vowel positions from 8: $^8C_3$

Number of ways to arrange the remaining 5 consonants: $5!$

Total number of valid arrangements: $^8C_3 \times 5! = \dfrac{8!}{3!}$

Correct answer is (D): $\dfrac{8!}{3!}$

Answer 2

Given:
Word is VERTICAL, which has 8 letters.

Vowels in the word: E, I, A (3 vowels) 

Step 1: Choose positions for the 3 vowels out of 8 letters: $^8C_3$

Step 2: Arrange the remaining 5 consonants in those 5 positions: $5!$

Total number of ways: $^8C_3 \times 5! = \dfrac{8!}{3!}$

Correct option is (D): $\dfrac{8!}{3!}$