1.

A candidate is required to answer 7 questions out of 12 questions, which are divided into two groups, each containing 6 questions. He is not permitted to attempt more than 5 questions from either group. He can choose the seven questions in 650 ways.

Answer» Since, condidate cannot attempt more than 5 questions from either group.
Thus, he is able to attempt minimum two questions from either group.
The number of questions attempted from each group is given in following table.
`{:("Group I ", 5, 4, 3, 2),("Group II ", 2, 3, 4, 5):}`
Since, each group have 6 questions and total attempted 7 questions.
`therefore" "` Total number of possible ways `= ""^(6)C_(5) xx ""^(6)C_(2)+ ""^(6)C_(4) xx ""^(6)C_(3) + ""^(6)C_(3) xx ""^(6)C_(4) + ""^(6)C_(2) xx ""^(6)C_(5)`
`= 2[""^(6)C_(5) xx ""^(6)C_(2) + ""^(6)C_(4) xx ""^(6)C_(3)]`
`= 2 [6 xx 15 + 15 xx 20]`
`= 2 [90 + 300]`
`= 2 xx 390 = 780`


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