Question

A student is given a true-false exam. with 10 questions. If he gets 8 or more correct answers he passes the exam. Given that he guesses at the answer to each question, the probability that he passes the exam is $k/128$. Find the value of $k$.

Answer 1

$n=$ total no. of ways $=2^{10}=1024.$
Since each answer can be true or false

and $m=$ favourable number of ways ${=}^{10}{C}_8{+}^{10}{C}_9{+}^{10}{C}_{10}=45+10+1=56$.
Since to pass the exam, he must give $8$ or $9$ or $10$ true answers,
Hence $P=\frac{m}{n}=\frac{56}{1024}=\frac{7}{128}$

$\therefore k=7$