6. From $m$ boys and $n$ girls $(10 \geqslant m>n \geqslant 4)$, 2 people are randomly selected to be class leaders. Let event $A$ represent the selection of 2 people of the same gender, and event $B$ represent the selection of 2 people of different genders. If the probability of $A$ is the same as the probability of $B$, then the possible values of $(m, n)$ are $\qquad$ .
According to the problem, $C_{m}^{2}+C_{n}^{2}=m n \Rightarrow m^{2}+n^{2}-(m+n)=2 m n \Rightarrow(m-n)^{2}=m+n$. In the set $\{4,5,6,7,8,9,10\}$, the pairs that satisfy $n+m$ being a perfect square are $4+5,6+10,7+9$. After verification, only $(m, n)=(10,6)$ meets the condition.