Question

In an examination, a questions paper consists of 12 questions divided into two parts i.e., part I and part II containing 5 and 7 questions, respectively. A student is required to attempt 8 questions in all, selecting atleast 3 from each part. The number of ways student can select the questions are

• A. 360
• B. 600
• C. 400
• D. 420

Answer 1

Answer: (D)

Here, we have to attempt 8 questions keeping in mind that atleast 3 from each part, then following possibility arises:

Hence, total number of ways
$=\left({ }^5 C_3 \times{ }^7 C_5\right)+\left({ }^5 C_4 \times{ }^7 C_4\right)+\left({ }^5 C_5 \times{ }^7 C_3\right) $
$ =\left({ }^5 C_2 \times{ }^7 C_2\right)+\left({ }^5 C_1 \times{ }^7 C_3\right)+\left({ }^5 C_5 \times{ }^7 C_3\right) \left(\because{ }^n C_r={ }^n C_{n-r}\right) $
$=\left(\frac{5 \times 4}{2} \times \frac{7 \times 6}{2}\right)+\left(5 \times \frac{7 \times 6 \times 5}{6}\right)+\left(1 \times \frac{7 \times 6 \times 5}{6}\right) $
$ =(10 \times 21)+175+35$
$=210+210$
$ =420 $ ways