Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is :

Solution

Maths sir · 4 steps

Fix M in the fourth slot, arrange the other 10 letters, and the repeat factors cancel out.

1. Count the letters. EXAMINATION has 11 letters. A, I and N each appear twice, and M appears only once.
2. Total words. All arrangements of the 11 letters number 11!/(2! 2! 2!). Each 2! removes the double counting from swapping two identical letters.
3. Favourable words. Put M in the fourth position. The remaining 10 letters still contain two A, two I and two N, so they can be arranged in 10!/(2! 2! 2!) ways.

   P = (10!/(2! 2! 2!))/(11!/(2! 2! 2!)) = 10!/11! = 1/11

4. So. The probability is 1/11, option (b). This also makes sense directly: the single M is equally likely to land in any of the 11 positions.

The idea to keep. For a letter that appears once, the chance it lands in a given position is 1 divided by the total number of letters. The repeated letters cancel in the ratio, so you do not need them.

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