Question

In an examination, a question 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 attempts 8 questions in all, selecting at least 3 from each part. In how many ways can a student select the questions ?

Answer 1

Here, part I has 5 questions and part II has 7 questions.

Student has to attempt 8 questions selecting at least 3 from each section.

So, Number of ways to select at least 3 from each section and a total of 8 questions. = (3 from part I and 5 from part II) or (4 from part I and 4 from part II) or (4 from part I and 4 from part II) or (5 from part I and 3 from part II)

$={(}^5{C}_3{\times }^7{C}_5)+{(}^5{C}_4{\times }^7{C}_5)+{(}^5{C}_3{\times }^7{C}_3)$

$=(\frac{5\times 4}{2\times 1}\times \frac{7\times 6}{2\times 1})+(5\times \frac{7\times 6\times 5}{3\times 2})+(1\times 7\frac{5\times 6\times 5}{3\times 2})$

$=210+175+35=420$

Required number of ways = 420.