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

Given :
Word is VERTICAL.
In this word, 3 three vowels are there i.e E, I, A.
Number of ways that out of 8 alphabets
3 vowels (EIA) can be chosen are ⁸C₃
and remaining 5 letters can be arranged in 5 ! ways.
Therefore, the number of ways :
\(={^8C_3}\times5!=\frac{8!}{3!}\)
So, the correct option is (D) : \(\frac{8!}{3!}\)

Answer 2

1. The word 'VERTICAL' has 8 letters.
2. It has 3 vowels: E, I, and A.
3. We want to arrange the letters without changing the order of the vowels, which means we only need to consider the consonants: V, R, T, C, and L.

Now, we have 5 consonants and 8 total letters. The number of arrangements of these 5 consonants among themselves is 5!, which is the number of permutations of the consonants.

However, we must account for the fact that the consonants are not distinct (there are repeated letters). The letter 'T' appears twice, so we divide the total arrangements by 2! for the repetition of 'T'. Similarly, the letter 'C' appears twice, so we divide by another 2!.

So, the total number of arrangements of the consonants is \(\frac{5!}{(2! \times 2!)}\).

The final answer is the number of arrangements of consonants multiplied by the number of arrangements of vowels (which is just 1 since we're not changing the order of vowels).

So, the total number of arrangements =\( (\frac{5!}{(2! \times 2!)}) * 1 = \frac{3!}{8!}.\)

This justifies why the correct answer is \(\frac{8!}{3!}\).

The correct answer is option (D): \(\frac{8!}{3!}\)