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Q.
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
Probability
Solution:
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 $\left({ }^{7} C_{2}\right)(14)=(21)(14)$. Thus, the required probability is $(21)(14) / 7^{4}=6 / 49$.