Question

The number of ways in which the letters of the word "BUSINESS" can be arranged so that the vowels always come together is

• A. 5!
• B. 6!
• C. 7!
• D. 8!

Hint:

When asked to arrange letters with some specific condition, treat the restricted group as a single block and then calculate the total arrangements.

Answer 1

The Correct Option is C

Step 1: Treating vowels as a block.
The word "BUSINESS" contains the vowels U, I, and E. To ensure that the vowels always come together, we treat them as a single unit or block. So, we have the following letters left: B, S, N, S, and the block of vowels (UIE).
Step 2: Arranging the blocks.
Now, we have 6 units: B, S, N, S, and the vowel block. The number of ways to arrange these 6 units is \( \frac{6!}{2!} \), because the letter S is repeated twice.
Step 3: Arranging the vowels.
The 3 vowels (UIE) can be arranged among themselves in \( 3! \) ways.
Step 4: Total number of arrangements.
The total number of ways to arrange the letters so that the vowels are together is: \[ \frac{6!}{2!} \times 3! = \frac{720}{2} \times 6 = 7! = 5040 \]
Step 5: Conclusion.
Therefore, the correct answer is (3) 7!.