Question

In an examination, 5 students have been allotted their seats as per their roll numbers. The number of ways, in which none of the students sit on the allotted seat, is:

Hint:

Derangements are a specific type of permutation where no object appears in its original position. Use the derangement formula to find solutions to such problems.

Answer 1

Correct Answer: 44

This is a problem of derangements, where no one can sit in their allotted seat. The formula for the number of derangements \( D_n \) of \( n \) objects is: \[ D_n = n! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \cdots + (-1)^n \frac{1}{n!} \right) \] For \( n = 5 \), the number of derangements is: \[ D_5 = 5! \left( 1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!} \right) \] \[ D_5 = 120 \left( 1 - 1 + \frac{1}{2} - \frac{1}{6} + \frac{1}{24} - \frac{1}{120} \right) \] \[ D_5 = 120 \times \frac{44}{120} = 44 \] Thus, the correct answer is \( \boxed{44} \).