A candidate is required to answer $6$ out of $12$ questions which are divided into two parts $A$ and $B$ each containing $6$ questions and he/she is not permitted to attempt more than $4$ questions from any part. In how many different ways can he/she make up his/her choice of $6$ questions?

The correct option is A $850$
The selections can be made in either of these three ways:
$i)2$ questions from section $A$ and $4$ from section $B$
$ii)3$ questions from each section
$iii)4$ questions from section $A$ and $2$ from section $B$
For case $i)$ we have ${}^6{C}_2{\times }^6{C}_4$ ways
For case $ii)$ we have ${}^6{C}_3{\times }^6{C}_3$ ways
For case $iii)$ we have ${}^6{C}_4{\times }^6{C}_2$ ways

Total number of ways $=225+400+225=850$