Step 1: Spot the distribution.
Each case is either solved or not, with a fixed chance, and there are 6 independent cases. This is a binomial distribution.
Step 2: Write the probabilities.
Chance solved $p = 25\% = \tfrac{1}{4}$. Chance not solved $q = 1 - \tfrac14 = \tfrac34$. Number of cases $n = 6$.
Step 3: Recall the formula.
\[ P(X = x) = \binom{n}{x} p^x q^{n-x} \]
We want at least 5 solved, so $P(X = 5) + P(X = 6)$.
Step 4: Find $P(X = 5)$.
\[ P(X=5) = \binom{6}{5}\left(\frac14\right)^5\left(\frac34\right)^1 = 6 \times \frac{1}{1024} \times \frac{3}{4} = \frac{18}{4096} \]
Step 5: Find $P(X = 6)$.
\[ P(X=6) = \binom{6}{6}\left(\frac14\right)^6 = 1 \times \frac{1}{4096} = \frac{1}{4096} \]
Step 6: Add them.
\[ P(X \ge 5) = \frac{18}{4096} + \frac{1}{4096} = \frac{19}{4096} \]
\[ \boxed{\dfrac{19}{4096}} \]