In an examination, 5 students have been allotted their seats as per their roll numbers.

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 sits on the allotted seat, is \_\_\_\_\_\_\_\_\_.

Answer: 44

Accepted Answer

This problem can be solved using the concept of derangements, which is a permutation of objects where no object appears in its original position. In this case, we have 5 students who should not sit in their allotted seats.

The formula for calculating the number of derangements (also known as subfactorials) is :

D(n) $=n![\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\ldots \ldots ..+(-1{)}^{n}\frac{1}{n!}]$

Where n is the number of students, in this case, 5.

Using the formula, let's calculate the derangements for 5 students :

D(5) = $5!(\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!})$

D(5) = $120(1-1+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120})$

D(5) = $120(0+\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120})$

D(5) = $120(\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120})$

D(5) = $120(\frac{60}{120}-\frac{20}{120}+\frac{5}{120}-\frac{1}{120})$

D(5) = $120(\frac{44}{120})$

D(5) = 44

So, there are 44 ways in which none of the students sits on the allotted seat.

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