The average marks of $13$ students is $67$. If the average marks of the first seven students is $61$ and the average marks of the last seven students is $65$, then what is the marks of the seventh student?

Let the marks of the seventh student be $x$.Total marks of all $13$ students = $13\times 67=871$ Total marks of the first $7$ students = $7\times 61=427$ Total marks of the last $7$ students = $7\times 65=455$ In adding the marks of the first seven and the last seven students, the seventh student's marks are counted twice:
$$\text{Sum of first 7}+\text{Sum of last 7}=\text{Total Sum of 13}+x$$
$$427+455=871+x$$
$$882=871+x$$
$$x=882-871=11$$
Therefore, the marks of the seventh student is $11$.