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. \(30240 \)

Hint:

Always check for repetitions twice: once when arranging the main units and again when arranging the internal block!

Answer 1

The Correct Option is D

Concept: To arrange items where certain elements must be together, treat the group of elements as a single "block" or "unit." Remember to divide by the factorial of the number of repetitions for any identical letters.
Step 1: Identify the vowels and consonants in 'ESPECIALLY'.
Total letters = 10.
Vowels: E, E, I, A (4 letters).
Consonants: S, P, C, L, L, Y (6 letters).

Step 2: Treat the 4 vowels as one single unit.
New total units = 6 (consonants) + 1 (vowel block) = 7 units.
In these 7 units, 'L' is repeated twice.
Arrangements of units = \(\frac{7!}{2!} = \frac{5040}{2} = 2520\).

Step 3: Arrange the vowels within their block.
Vowels are E, E, I, A. There are 4 letters with 'E' repeated twice.
Internal arrangements = \(\frac{4!}{2!} = \frac{24}{2} = 12\).

Step 4: Calculate total arrangements.
Total ways = \(2520 \times 12 = 30240\).