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 each group. The number of ways in which he can choose the $7$ questions is

Answer 1

A candidate can attempt $5$ questions from group $I$ and $2$ from group $II$ or $4$ from group $I$ and $3$ from group $II$ or $3$ from group $I$ and $4$ from group $II$ or $2$ from group $I$ and $5$ from group $II$.

This can be done in

${}^6{C}_5{\times }^6{C}_2{+}^6{C}_4{\times }^6{C}_3{+}^6{C}_3{\times }^6{C}_4{+}^6{C}_2{\times }^6{C}_5=6\times 15+15\times 20+20\times 15+15\times 6=90+300+300+90=780$