Question

If the best and the worst papers never appear together, then six examination papers can be arranged in how many ways?
A) 120
B) 480
C) 240
D) None of these

Answer 1

Hint: We have to find all the number of ways in which six examination papers can be arranged and the number of ways in which best and worst paper appears together and then we will subtract the number of ways in which best and worst paper appears together from the number of ways six examination papers can be arranged.

Complete step by step answer:
Total cases in which six examination papers can be arranged \[ = 6!\]
So, when the worst and best paper comes together then they can be considered as one possibility, and the remaining will be $5!$ and both papers are interchangeable from their places so we will multiply $5!$ with 2. So,
Total cases in which the best and worst papers appear together $ = 5! \times 2$
Total cases in which the best and worst papers never appear together $=$ Total cases in which six examination papers can be arranged $–$ Total cases in which the best and worst papers appear together.
 $ = 6! - 5! \times 2$
 $ = 720 - 240$
 $ = 480$
Therefore, there are 480 ways in which 6 examination papers can be arranged where the worst and best papers never appear together. So, option (B) is correct.

Note:
The concepts of permutation and combination are used to find the total number of cases and then subtract it with the non favourable number of cases so we can get the total number of favourable cases. In these types of questions, it is often in students to forget to multiply 5! with 2, so we must be little aware of it.