The number of words, with or without meaning, that can be formed by taking 4 letters at a time from the letters of the word ’SYLLABUS’ such that two letters are distinct and two letters are alike, is :

Solution

Maths sir · 4 steps

Pick the doubled letter, then the two single letters, then arrange: three clean stages.

1. Sort the letters. SYLLABUS has 8 letters: S twice, L twice, and Y, A, B, U once each. So there are 6 different letters, and only S and L can appear as a pair.
2. Choose the letters. The alike pair is SS or LL, which gives 2 ways. The two distinct letters come from the 5 letters left, and that includes the other repeated letter used once. That gives C(5, 2) = 10 ways.
3. Arrange each choice. Each choice has 4 letters with one letter repeated twice, so it can be arranged in 4!/2! = 12 ways.

   2 × C(5, 2) × 4!/2! = 2 × 10 × 12 = 240

4. So. The number of words is 240.

The common slip: 144. Choosing the two distinct letters only from Y, A, B, U gives 2 × 6 × 12 = 144. The other repeated letter can also be used once, so you choose from 5 letters, not 4.

The idea to keep. With repeated letters, select first and arrange later. Count how many letters can actually form a pair, then divide by the factorial of each repeat when you arrange.

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