A student has to answer 10 out of 13 questions in an examination choosing atleast 5 questions from the first…
A student has to answer 10 out of 13 questions in an examination choosing atleast 5 questions from the first 6 questions. The number of choice available to the student is
$63$
$91$
$161$
$196$
Solution
There are two cases arise.
Case I When 5 questions are selected from first 6 questions and next 5 questions are selected from 7 questions.
$\begin{aligned} \therefore \text { Number of ways } & ={ }^6 C_5 \times{ }^7 C_5 \\ & =6 \times \frac{7 \times 6}{2 \times 1} \\ & =126\end{aligned}$
Case II When 6 questions are selected from first 6 questions and next 4 questions are selected from 7 questions.
$\begin{aligned} \therefore \text { Number of ways } & ={ }^6 C_6 \times{ }^7 C_4 \\ & =1 \times \frac{7 \times 6 \times 5}{3 \times 2}=35\end{aligned}$
$\therefore$ Required number of ways $=126+35=161$