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

Answer 1

It is given that the question paper consists of 12 questions divided into two parts - Part I and Part II, containing 5 and 7 questions, respectively.
A student has to attempt 8 questions, selecting at least 3 from each part.

This can be done as follows.
(a) 3 questions from part I and 5 questions from part II
(b) 4 questions from part I and 4 questions from part II
(c) 5 questions from part I and 3 questions from part II

3 questions from part I and 5 questions from part II can be selected in $^5C_3\times\space^7C_5$ ways.
4 questions from part I and 4 questions from part II can be selected in $^5C_4 \times\space^7C_4$ ways.
4 questions from part I and 4 questions from part II can be selected in $^5C_5\times\space^7C_3$ ways.
Thus, required number of ways of selecting questions
=$$^5C_3\times\space^7C_5+^5C_4\times\space^7C_4+^5C_5\times\space^7C_3

=$$\frac{5!}{2!3!}\times\frac{7!}{2!5!}+\frac{5!}{4!1!}\times\frac{7!}{4!3!}+\frac{5!}{5!0!}\times\frac{7!}{3!4!}

$=210+175+35=420$