Question

A test consists of $6$ multiple choice questions, each having $4$ alternative answers of which only one is correct. The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct, is62ac68167b6373095ac09e9e

Answer 1

Finding the number of ways in which a candidate answers all six questions such that exactly four of the answers are correct:

Given data: A test contains $6$ multiple choice questions, each having $4$ alternative answers of which only one is correct.

Ways of selecting correct questions $={}^{6}C_4$

Way of doing them correct $=1$

Ways of doing remaining $2$ questions incorrect $=3^2$ [Three wrong options out of four]

$\therefore$The number of ways, in which a candidate answers all six questions such that exactly four of the answers are correct is:

$\begin{array}{rcl}{}^{6}C_4\times 1\times 3^2& =& \frac{6!}{2!4!}\times 9\\ & =& \frac{6\times 5\times 4!}{2\times 4!}\times 9\\ & =& \frac{30}{2}\times 9\\ & =& 135\end{array}$

Hence, the number of ways is $\mathbf{135}$