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7 students test results and grade sheets were in the hands of a careless teacher. For each of the 7 students, he entered someone else's grades. How many ways could this mistake have been made by him?

Option: 1

1854


Option: 2

5040


Option: 3

3186

 


Option: 4

4340


Answers (1)

best_answer

 

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

\begin{aligned} & D_7=7 ! \times\left(\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\frac{1}{5 !}+\frac{1}{6 !}-\frac{1}{7 !}\right) \\ & D_5=5 ! \times\left(\frac{1}{2 !}-\frac{1}{3 !}+\frac{1}{4 !}-\frac{1}{5 !}\right) \\ & D_7=7 ! \times\left(\frac{1854}{5040}\right)_{\mid} \\ & D_7=1854 \end{aligned}

The teacher could have made this error in 1854 different ways.

The total number of ways in which 7 mark sheets can be given to 7 students in 7! = 5040 ways.

Of these, in 1854 ways, all the marks entered would be incorrect.

So, the number of ways in which he could have entered correct marks for at least one student = 5040 – 1854 = 3186. 

Therefore, the number of ways is 3186.

Posted by

HARSH KANKARIA

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