A quiz has 10 questions, each with four options and exactly one right option…

Step 1. Exactly three. $\binom{10}{3}\left(\dfrac{1}{4}\right)^3\left(\dfrac{3}{4}\right)^7 = 120 \times \dfrac{2187}{4^{10}} = \dfrac{262\,440}{1\,048\,576} \approx 0.2503$.

Step 2. At least six. The terms for $6, 7, 8, 9, 10$ are about $0.01622$, $0.00309$, $0.00039$, $0.00003$ and $0.000001$. They add up to about $0.0197$.

Step 3. The standard deviation. $\sqrt{npq} = \sqrt{10 \times 0.25 \times 0.75} = \sqrt{1.875} \approx 1.3693$.

What it means. A student who guesses is right about $2$ or $3$ times in $10$, and gets $6$ or more right about once in $50$ quizzes.