Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$.  What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number?

$\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$
To determine the probability that \((\cos(a\pi) + i\sin(b\pi))^4\) is a real number, we need to analyze the conditions under which this expression is real. 

1. **Expression Simplification**:
   \[
   (\cos(a\pi) + i\sin(b\pi))^4
   \]
   Using De Moivre's Theorem, we know:
   \[
   (\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)
   \]
   Therefore:
   \[
   (\cos(a\pi) + i\sin(b\pi))^4 = \cos(4a\pi) + i\sin(4a\pi)
   \]
   For this to be a real number, the imaginary part must be zero:
   \[
   \sin(4a\pi) = 0
   \]
   This implies:
   \[
   4a\pi = k\pi \quad \text{for some integer } k
   \]
   Simplifying, we get:
   \[
   4a = k \quad \Rightarrow \quad a = \frac{k}{4}
   \]
   Since \(a\) is in the interval \([0, 2)\), \(k\) can be \(0, 1, 2, 3, 4, 5, 6, 7\).

2. **Conditions for \(b\)**:
   Similarly, for \((\cos(a\pi) + i\sin(b\pi))^4\) to be real, we need:
   \[
   \cos(4b\pi) = \pm 1
   \]
   This implies:
   \[
   4b\pi = m\pi \quad \text{for some integer } m
   \]
   Simplifying, we get:
   \[
   4b = m \quad \Rightarrow \quad b = \frac{m}{4}
   \]
   Since \(b\) is in the interval \([0, 2)\), \(m\) can be \(0, 1, 2, 3, 4, 5, 6, 7\).

3. **Counting Rational Numbers**:
   We need to count the rational numbers \(a\) and \(b\) in the interval \([0, 2)\) that can be written as \(\frac{n}{d}\) where \(1 \leq d \leq 5\).

   - For \(d = 1\): \(a\) and \(b\) can be \(0, 1\) (2 values each).
   - For \(d = 2\): \(a\) and \(b\) can be \(0, \frac{1}{2}, 1, \frac{3}{2}\) (4 values each).
   - For \(d = 3\): \(a\) and \(b\) can be \(0, \frac{1}{3}, \frac{2}{3}, 1, \frac{4}{3}, \frac{5}{3}\) (6 values each).
   - For \(d = 4\): \(a\) and \(b\) can be \(0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, 1, \frac{5}{4}, \frac{3}{2}, \frac{7}{4}\) (8 values each).
   - For \(d = 5\): \(a\) and \(b\) can be \(0, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, 1, \frac{6}{5}, \frac{7}{5}, \frac{8}{5}, \frac{9}{5}\) (10 values each).

4. **Total Number of Rational Numbers**:
   Summing these, we get:
   \[
   2 + 4 + 6 + 8 + 10 = 30 \quad \text{values for each of } a \text{ and } b
   \]
   Therefore, there are \(30 \times 30 = 900\) possible pairs \((a, b)\).

5. **Valid Pairs**:
   We need to count the pairs \((a, b)\) such that \(a = \frac{k}{4}\) and \(b = \frac{m}{4}\) for \(k, m \in \{0, 1, 2, 3, 4, 5, 6, 7\}\). Each \(a\) and \(b\) has 8 possible values, so there are \(8 \times 8 = 64\) valid pairs.

6. **Probability Calculation**:
   The probability is:
   \[
   \frac{64}{900} = \frac{16}{225} = \frac{6}{25}
   \]

The final answer is \(\boxed{\frac{6}{25}}\)