24. For any rational numbers $a, b \in [0,2)$, when $a, b$ are expressed in the form $\frac{n}{d} (n, d \in \mathbf{Z})$, $1 \leqslant d \leqslant 5$. Then the probability of the ordered pairs $(a, b)$ such that $(\cos a \pi + \mathrm{i} \sin b \pi)^{4}$ is a real number is ( ).
(A) $\frac{3}{5}$
(B) $\frac{4}{25}$
(C) $\frac{41}{200}$
(D) $\frac{6}{25}$
(E) $\frac{13}{50}$
24. D.

Given
$$
\begin{array}{r}
a, b \in \left\{0,1, \frac{1}{2}, \frac{3}{2}, \frac{1}{3}, \frac{2}{3}, \frac{4}{3}, \frac{5}{3}, \frac{1}{4}, \frac{3}{4}, \right. \\
\left. \frac{5}{4}, \frac{7}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{6}{5}, \frac{7}{5}, \frac{8}{5}, \frac{9}{5}\right\} .
\end{array}
$$

Thus, the ordered pairs $(a, b)$ have $20 \times 20$ possible scenarios.
Let $x = \cos a \pi, y = \sin b \pi$. Then
$$
\begin{array}{l}
(\cos a \pi + \mathrm{i} \sin b \pi)^{4} = (x + y \mathrm{i})^{4} \\
= x^{4} + 4 x^{3} y \mathrm{i} - 6 x^{2} y^{2} - 4 x y^{3} \mathrm{i} + y^{4} \\
= \left(x^{4} - 6 x^{2} y^{2} + y^{4}\right) + 4 x y \left(x^{2} - y^{2}\right) \mathrm{i} .
\end{array}
$$

Since $(\cos a \pi + \mathrm{i} \sin b \pi)^{4}$ is a real number, we have $x y \left(x^{2} - y^{2}\right) = 0$.
【Case 1】When $x = 0$ or $y = 0$, i.e., $\cos a \pi = 0$ or $\sin b \pi = 0$, we have
$$
a = \frac{1}{2}, \frac{3}{2}; b = 0, 1.
$$

Then the probability that satisfies the condition is
$$
P_{1} = 1 - \frac{18 \times 18}{20 \times 20} = \frac{76}{400}.
$$

【Case 2】When $x^{2} - y^{2} = 0$ and $x y \neq 0$, i.e.,
$$
\cos a \pi = \pm \sin b \pi \neq 0
$$

then either $a \in \{0, 1\}, b \in \left\{\frac{1}{2}, \frac{3}{2}\right\}$, which has $2 \times 2$ possible scenarios; or $a \in \left\{\frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}\right\}$, in which case $b \in \left\{\frac{1}{4}, \frac{3}{4}, \frac{5}{4}, \frac{7}{4}\right\}$, which has $4 \times 4$ possible scenarios.
Then the probability that satisfies the condition is
$$
P_{2} = \frac{2 \times 2 + 4 \times 4}{20 \times 20} = \frac{20}{400}.
$$

In summary, $P = \frac{76 + 20}{400} = \frac{6}{25}$.