A candidate is required to answer 7 questions out of 12 questions which are divided into two groups, each containing 6 questions.
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?
Solution:
Required ways $={ }^{6} C_{5} \times{ }^{6} C_{2}+{ }^{6} C_{4} \times{ }^{6} C_{3}+{ }^{6} C_{3} \times{ }^{6} C_{4}+{ }^{6} C_{2} \times{ }^{6} C_{5}$
$=2\left({ }^{6} C_{5} \times{ }^{6} C_{2}+{ }^{6} C_{4} \times{ }^{6} C_{3}\right)$
$=2(90+300)$
$=2(390)$
$=780$