Don't cheat!

Level pending

6 students were doing a small exam. Afterwards, they exchanged papers with each other to mark. The teacher said: " Don't cheat, you can't mark your own exam!" How many ways are there for the students to exchange their papers? (Assume the students are honest and they don't mark their own papers)


The answer is 265.

Takeda Shigenori by Takeda Shigenori , 46, Malaysia

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1 solution

Kenneth Tan
Mar 14, 2015

Relevant wiki: Derangements

This is a derangement problem, the formula for the number of derangements is D n = n ! ∑ i = 0 n ( − 1 ) i i ! \displaystyle D_n=n!\sum_{i=0}^n \frac{(-1)^i}{i!}

Set n = 6 n=6 , we get D 6 = 6 ! ( 1 0 ! − 1 1 ! + 1 2 ! − 1 3 ! + 1 4 ! − 1 5 ! + 1 6 ! ) = 265 \begin{aligned} D_6&=6!\left(\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}\right)\\&=265\end{aligned}

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