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