Question

In an examination of $10$ multiple choice questions ($1$ or more can be correct out of $4$ options), A student decides to mark the answers at random, Find the probability that he gets exactly two questions correct.

Answer 1

A student can mark $15$ different answers to a MCQ with $4$ option i.e. ${}^4{\text{C}}_1{+}^4{\text{C}}_2{+}^4{\text{C}}_3{+}^4{\text{C}}_4=15$
Hence if he marks the answer at random, chance that his answer is correct$=\frac{1}{51}$ and being incorrecting $\frac{14}{15}.$ Thus $p=\frac{1}{15},q=\frac{14}{15}.$
$P$(2 success)${=}^{10}{\text{C}}_2\times {\left(\frac{1}{15}\right)}^2\times {\left(\frac{14}{15}\right)}^8$