Question

In how many different ways, can the letters of the words EXTRA be arranged so that the vowels are never together?

• A. 48
• B. 120
• C. 72
• D. 168

Answer 1

The Correct Option is C

To determine the number of different ways the letters of the word "EXTRA" can be arranged so that the vowels are never together, we need to follow these steps:

1. Identify the vowels and consonants in the word "EXTRA". In this case, the vowels are E and A, and the consonants are X, T, and R.
2. First, consider the arrangement of the consonants. The 3 consonants, X, T, and R, can be arranged in \(3!\) ways.
   \(3! = 3 \times 2 \times 1 = 6\) ways. 
3. Create gaps between these consonants to place the vowels: "_X_T_R_". This creates 4 possible slots (before X, between X and T, between T and R, and after R) for placing the vowels.
4. Now, the problem reduces to selecting 2 out of these 4 gaps to place the vowels, ensuring that they are not together.
5. The number of ways to select 2 slots out of the 4 available is calculated as \(\binom{4}{2}\).
   \(\binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6\) ways.
6. For each selection of slots, the 2 vowels E and A can be arranged within them in \(2!\) ways.
   \(2! = 2 \times 1 = 2\) ways.
7. Therefore, combining these arrangements, the total number of ways to arrange the letters so that the vowels are not together is calculated as follows:
   \(3! \times \binom{4}{2} \times 2! = 6 \times 6 \times 2 = 72\) ways.

Therefore, the correct answer is 72.