3. (D)

The polar equation of a conic section $\rho=\frac{p}{1-e \cos \theta}$,
represents a hyperbola if and only if $e>1$. Therefore, the equation in this problem must satisfy
$$
C_{m}^{n}>1 \text {, }
$$

Given that
$$
1 \leqslant n \leqslant m \leqslant 5,
$$

we can obtain 6 different values:
$$
C_{3}^{1}, C_{3}^{2}, C_{4}^{1}, C_{4}^{2}, C_{5}^{1}, C_{5}^{2} .
$$

These correspond to 6 different hyperbolas.
3. For all $m, n$ satisfying $1 \leqslant n \leqslant m \leqslant 5$, the polar equation
$$
\rho=\frac{1}{1-C_{m}^{n} \cos \theta}
$$

represents the number of different hyperbolas is
(A) 15 ;
(B) 10 ;
(C) 7 ;
(D) 6 .