2. Roll two fair six-sided dice numbered $1 \sim 6$ (integers) with one red and one green. The numbers facing up on the red and green dice are used as the coefficients of the linear term and the constant term, respectively, in the quadratic function $y=x^{2}+m x+n$. The probability that the quadratic function $y=x^{2}+m x+n$ intersects the $x$-axis at two different points is ( ).
(A) $\frac{5}{12}$
(B) $\frac{4}{9}$
(C) $\frac{17}{36}$
(D) $\frac{1}{2}$
2.C.

By operation, 36 quadratic functions can be obtained. According to the problem, $\Delta=m^{2}-4 n>0$, which means $m^{2}>4 n$. Through enumeration, it is known that there are 17 pairs of $m, n$ that satisfy the condition. Therefore, $P=\frac{17}{36}$.