Question

If the papers of $4$ students can be checked by any one of the $7$ teachers, then the probability that all the $4$ papers are checked by exactly $2$ teachers is

• A. $\frac{2}{7}$
• B. $\frac{12}{49}$
• C. $\frac{32}{343}$
• D. None of these

Answer 1

The correct option is D None of these

The total number of ways in which papers of $4$ students can be checked by seven teachers is $7^4$. The number of ways of choosing two teachers out of $7$ is ${}^7{C}_2$. The number of ways in which they can check four papers is $2^4$. But this includes two ways in which all the papers will be checked by a single teacher. Therefore, the number of ways in which $4$ papers can be checked by exactly two teachers is $2^4-2=14$. Therefore, the number of favorable ways is ${(}^7{C}_2\left)\right(14)=21\times 14$. Thus, the required probability is $\frac{21\times 14}{7^4}=\frac{6}{49}$.