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

Explanation

`:'` The question paper is divided into two parts I and II, containing 5 and 7 questions respectively. A student is required to select total 8 questions while at least 3 questions is required to select from each part, then the selection can be made in the following ways:
(i) 3 questions from part I and 5 questions from part II.
(ii) 4 questions from part I and 4 questions from part II.
(iii) 5 questions from part I and 3 questions from part II.
In first case, a student is required to select 3 questions out of 5 from part I and 5 questions out of 7 from part II, whose total no. of ways `= .^(5)C_(3) xx .^(7)C_(5)`
`=(lfloor5)/(lfloor3.ul(5-3)) xx (lfloor7)/(lfloor5.ul(7-5))`
`= (lfloor5)/(lfloor3.lfloor2) xx (lfloor7)/(lfloor5 lfloor2) = (lfloor7)/(lfloor3.lfloor2.lfloor2)`
`= (7 xx 6 xx 5 xx 4 xx lfloor3)/(2 xx 1 xx 2 xx 1 xx lfloor3) = 210`
In second case, a student is required to select four-four questions from each part, whose total no. of ways
`= .^(5)C_(4) xx .^(7)C_(4)`
`=(lfloor5)/(lfloor4.ul(5-4)) xx(lfloor7)/(lfloor4.ul(7-4)) =(lfloor5)/(lfloor4lfloor1) xx (lfloor7)/(lfloor4lfloor3)`
`= (5xx lfloor4)/(1xx lfloor4) xx (7xx 6xx 5 xx lfloor4)/(3 xx 2 xx 1xx lfloor4) = 5 xx 35 = 175`
In third case, a student is required to select 5 questions from part I and 3 questions from part II, whose total no. of ways
`= .^(5)C_(5) xx .^(7)C_(3)`
`= 1 xx (lfloor7)/(lfloor3.ul(7-3)) = (lfloor7)/(lfloor3.lfloor4)`
`= 1 xx (7 xx 6 xx 5 xx lfloor4)/(3 xx 2 xx 1 xx lfloor4) = 35`
`:.` Total ways of selecting 8 questions
`= 210 +175 +35 = 420`
Therefore, a student can select the questions in 420 ways.