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  • A teacher mistakenly distributes 5 textbooks to 5 students in a classroom, but each student receives a textbook intended for a different student. In how many ways could this distribution

A teacher mistakenly distributes 5 textbooks to 5 students in a classroom, but each student receives a textbook intended for a different student. In how many ways could this distribution mistake have occurred?

Option: 1

54


Option: 2

66


Option: 3

76


Option: 4

92


Answers (1)

Applying the derangement formula, the number of ways in which this error could have been done is given by,

\begin{aligned} & D_5=5 ! \times\left(\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\frac{1}{5 !}\right) \\ & D_5=120 \times\left(\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120}\right) \\ & D_5=120 \times\left(\frac{60-20+15-1}{120}\right) \\ & D_5=54 \end{aligned}

The teacher could have made this error in 54 different ways.

The total number of ways in which 5 books can be distributed to 5 students in 5! = 120 ways.

Of these, in 54 ways, all the books distributed would be incorrect.

So, the number of ways in which the teacher could have distributed correct books for at least one student = 120 – 54= 66. 

Therefore, the number of ways is 66.

 

Posted by

Sumit Saini

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