Question

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. In how many ways can he choose the 7 questions ?

Answer 1

Total number of questions = 12

Total number of questions to be answered = 7

Each group has 6 question (6+6) more than 5 question from either group is not perimetted, therefore the number of ways a student can choose questions can be done in following ways.

${}^6{C}_2{\times }^6{C}_5{\times }^6{C}_4{\times }^6{C}_4{\times }^6{C}_3{\times }^6{C}_3{\times }^6{C}_5{\times }^6{C}_2=2{(}^6{C}_2{\times }^6{C}_5{\times }^6{C}_3{\times }^6{C}_4)=2(\frac{6!}{2!4!}\times \frac{6!}{5!1!}+\frac{6!}{3!3!}\times \frac{6!}{4!2!})=2(\frac{6\times 5\times 6}{2}+\frac{6\times 5\times 4\times 6\times 5}{3\times 2\times 2})=\frac{2\times 6\times 5\times 6}{2}(1+\frac{20}{6})=180(\frac{26}{6})=30\times 26=780$
= 780 ways