If the papers of 4 students can be checked by any one of the seven teachers, then the probability that all the four papers are checked by exactly two teachers is

The total number of ways in which papers of 4 students can be checked by seven teacher is $7^4$

The number of ways of choosing two teachers out of 7 is ${}^7\mathrm{C}_2$. The no. 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.

= The number of ways in which 4 papers can be checked by exactly two teachers is $2^4-2=14$

$\therefore$ The number of favourable ways $=\left({}^7\mathrm{C}_2\right)\left(14\right)=\left(21\right)\left(14\right)$

$\therefore$ Required probability $=\frac{\left(21\right)\left(14\right)}{7^4}=\frac{6}{49}$