Probability Practice Test 1

The paper of each student can go to any of the nine professors.
As there are 5 students, there are 9 × 9 × 9 × 9 × 9 = 95
ways in which the papers can be checked by the professors.
Now, number of ways of selecting two professors ${=}^9{\mathit{C}}_2$
The five papers can be checked by the two professors in 25 ways, but this will contain two ways in which the papers are checked by just one professor.
∴ The number of ways in which 5 answer sheets are checked by exactly two professors ${=}^9{\mathit{C}}_2\times \left(2^5-2\right)$
Number of ways in which each paper can be checked by a professor = 2
∴ Number of ways such that five papers can be checked by those 2 professors ${=}^9{\mathit{C}}_2\times \left(2^5-2\right)$
Probability $=\frac{{}_{}^9{\mathit{C}}_2^{}\times \left(2^5-2\right)}{95}=\frac{40}{2187}$