4. For all $m, n$ satisfying $1 \leqslant n \leqslant m \leqslant 5$, the number of different hyperbolas represented by the polar equation $\rho=\frac{1}{1-\mathrm{C}_{m}^{n} \cos \theta}$ is ( ).
(A) 6
(B) 9
(C) 12
(D) 15
4. A.

When $m=n$, $\rho=\frac{1}{1-\mathrm{C}_{m}^{n} \cos \theta}$ represents a parabola.
When $m \neq n$, let $e=\mathrm{C}_{m}^{n}>1$.
Notice that,
$$
\mathrm{C}_{2}^{1}=\mathrm{C}_{4}^{2}, \mathrm{C}_{3}^{1}=\mathrm{C}_{3}^{2}, \mathrm{C}_{4}^{1}=\mathrm{C}_{4}^{3}, \mathrm{C}_{5}^{2}=\mathrm{C}_{5}^{3} \text {. }
$$

Therefore, the eccentricity $e>1$ has $10-4=6$ different values.