Question

In an examination, 5 students have been allotted their seats or per their roll numbers. The number of ways, in which none of the students sits on the allotted seat is 4

(A) 44
(B) 45
(C) 46
(D) 50​

Answer 1

The number of ways in which 5 students can be seated so that none of them sits in their allotted seat is 44, which is calculated using the derangement formula.

Step-by-step explanation:

The problem involves calculating the number of ways in which 5 students can be seated such that none of them sits in their own allotted seat.

This type of problem is a classical example of a derangement problem, also known as the hat-check problem.

To find the number of derangements (often represented by the symbol !n), we can use the formula for derangement which is [n!(1 - 1/1! + 1/2! - 1/3! + ... + (-1)^n/n!)] for n = 5. The calculation would give us a result of 44 ways in which none of the students sits in their allotted seat.

To perform the calculation, we would evaluate the expression: 5!(1 - 1/1! + 1/2! - 1/3! + 1/4! - 1/5!) which equals to 120(1 - 1 + 1/2 - 1/6 + 1/24 - 1/120) = 120(0 + 1/2 - 1/6 + 1/24 - 1/120) = 120(1/2 - 1/6 + 1/24 - 1/120)

= 120(30/60 - 10/60 + 2.5/60 - 1/120) = 120(20/60 + 2.5/60 - 1/120) = 120((120 + 5 - 1)/120) = 120(124/120)

= 44 ways.