Eight students from Think Academy School take a Mathematics test. Unfortunately, none of the students wrote his/her name on the test paper. As a result, the tests are handed back to the students at random. In how many ways can exactly $5$ of the $8$ students get the correct test back? 
Overseas Competition->Knowledge Point->Counting Modules->Permutations and Combinations->Combinations
If exactly $5$ pupils get the correct test, then exactly $3$ pupils must get the wrong test.  No. of ways to choose $5$ pupils to get the correct test is $$\frac{8 \times 7 \times 6 \times 5 \times 4}{5 \times 4 \times 3 \times 2 \times 1}-56.$$  To make sure that the other $3$ pupils get the wrong tests, the correct number is $2$.  Hence, the total no, of ways $=56 \times2=112$. 