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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?
Solution
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.
- 3 questions from part I and 5 questions from part II
- 4 questions from part I and 4 questions from part II
- 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 xx ""^7C_5`ways.
4 questions from part I and 4 questions from part II can be selected in `""^5C_4 xx ""^7C_4` ways.
5 questions from part I and 3 questions from part II can be selected in `""^5C_5 xx ""^7C_3` ways.
Thus, required number of ways of selecting questions
= 5C3 x 7C5 + 5C4 x 7C4 + 5C5 x 7C3
= `(5!)/(2!3!) xx (7!)/(2!5!) xx (5!)/(4!1!) xx (7!)/(4!3!) xx (5!)/(5!0!) xx (7!)/(3!4!)`
= `(5 xx 4)/(2) xx (7 xx 6)/(2) + 5 xx (7 xx 6 xx 5)/(3 xx 2) + 1 xx (7xx 6 xx 5)/(3 xx 2)`
= 210 + 175 + 35
= 420
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