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?
Given: 12 Qs are divided into 2 parts – Part I and Part II consisting of 5 and 7 Qs respectively. A student must attempt 8 Qs with atleast 3 from each part. This can be done as:
(a) 3 Qs from part I and 5 Qs from part II.
(b) 4 Qs from part I and 4 Qs from part II.
(c) 5 Qs from part I and 3 Qs from part II.
The first case can be selected in \( ^5C_3\times ^7C_5\) ways.
The second case can be selected in \( ^5C_4\times ^7C_4\) ways.
The third case can be selected in \( ^5C_5\times ^7C_3\) ways.
Thus, number of ways of selecting Qs
\( ^5C_3\times ^7C_5+ ^5C_4\times ^7C_4+ ^5C_5\times ^7C_3\)
= \( \dfrac{5!}{3!2!}\times \dfrac{7!}{2!5!}+\dfrac{5!}{1!4!}\times \dfrac{7!}{4!3!}+\dfrac{5!}{5!0!}\times \dfrac{7!}{3!4!}\)
= 210 +175 + 35 = 420
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