Question

Let n books be distributed to n students. Suppose that the books are returned and distributed to the students again.Later on in how many can the books be distributed so that no student will get the same book twice.

• A. $n!D_n$
• B. $D_n$
• C. $(n!{)}^2$
• D. n!

Answer 1

Answer: (C)

$(n!{)}^2$

Consider the problem

The first time the books are distributed $n!$ wayssince no student gets the same book that he got first time And the second time the books are distributed in $D_n$ ways.

Therefore, the total number of ways

$\begin{array}{c}n!D_n=n!n!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots +{\left(-1\right)}^n\frac{1}{n!}\right]\\ ={\left(n!\right)}^2\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots +{\left(-1\right)}^n\frac{1}{n!}\right]\end{array}$