The number of words (with or without meaning) that can be formed from all the letters of the word “LETTER” in which vowels never come together is ________ .

Solution

Maths sir · 4 steps

Seat the consonants first, then drop the two E's into separate gaps so they can never touch.

1. Sort the letters. LETTER has 6 letters. The vowels are E, E (two identical). The consonants are L, T, T, R, with T repeated.
2. Arrange consonants. L, T, T, R can be arranged in 4!/2! = 12 ways, dividing by 2! because the two T's are identical.
3. Place the vowels. Four consonants leave 5 gaps (both ends and the 3 between). Put the two E's in two different gaps so they stay apart. The E's are identical, so this is a choice, not an arrangement: C(5, 2) = 10 ways.

   4!/2! × C(5, 2) = 12 × 10 = 120

4. So. Check by subtraction: all words = 6!/(2! 2!) = 180, and words with EE together = 5!/2! = 60. Then 180 - 60 = 120, so the answer is 120.

The common slip: 150. Dividing by an extra 2! for the E's inside the EE block gives 5!/(2! 2!) = 30 together, and so 150. Two identical E's in a block have only one order.

The idea to keep. For letters that must never come together, arrange the others first and choose separate gaps for them. Divide by the factorial of every repeated letter, and only once.

Answer from the NTA final answer key. Solution by PrepWiser.