Question

In how many ways can the letters of the word 'PERMUTATIONS' be arranged if vowels are together?

• A. 4538400
• B. 20160
• C. 40320
• D. 2419200

Hint:

When solving permutation problems with repetition, account for the repeated letters by dividing by the factorial of their frequency.

Answer 1

The Correct Option is D

The word "PERMUTATIONS" has 11 letters in total. The vowels in "PERMUTATIONS" are E, U, A, I, O, which are 5 vowels. 
Step 1: Treat all the vowels as a single unit. This reduces the problem to arranging the 7 consonants and the unit of vowels. Thus, we have 7 + 1 = 8 units to arrange. The number of ways to arrange these 8 units is: \[ 8! = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 40320 \]
Step 2: Within the unit of vowels, the vowels can be arranged in \( 5! \) ways: \[ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \] 
Step 3: Since the letter "T" repeats twice in "PERMUTATIONS", we divide by \( 2! \) to account for the repetition: \[ \text{Total arrangements} = \frac{8! \times 5!}{2!} = \frac{40320 \times 120}{2} = 2419200 \] Thus, the total number of arrangements is 2419200.