Question

In how many ways can the word 'ESPECIALLY' be arranged such that the vowels always come together?

• A. 42600
• B. 25200
• C. 10080
• D. 30240
  (E) 29242
• E. 29242

Hint:

Don't forget to account for repetitions both in the main arrangement and inside the vowel block. If 'L' repeats in the consonants, divide the main part; if 'E' repeats in the vowels, divide the internal part.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem combines the "together" constraint with permutations of identical items. We must divide by the factorials of repeating letters to avoid overcounting.

Step 2: Key Formula or Approach:
Total = $\frac{(\text{Units})!}{\text{Repeating Consonants}!} \times \frac{(\text{Vowels})!}{\text{Repeating Vowels}!}$

Step 3: Detailed Explanation:
1. Word: ESPECIALLY (10 letters). 2. Vowels: E, E, I, A (4 letters). Consonants: S, P, C, L, L, Y (6 letters). 3. Treat vowels (EEIA) as 1 unit. Total units = 6 consonants + 1 unit = 7 units. 4. Arrange 7 units: Note that 'L' repeats twice. \[ \text{Ways} = \frac{7!}{2!} = \frac{5040}{2} = 2520 \] 5. Arrange vowels internally: (E, E, I, A). Note that 'E' repeats twice. \[ \text{Ways} = \frac{4!}{2!} = \frac{24}{2} = 12 \] 6. Total = $2520 \times 12 = 30240$.

Step 4: Final Answer:
The word can be arranged in 30240 ways.