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

Required ways = 

\[{}^6 C_5 \times^6 C_2 +^6 C_4 \times^6 C_3 + {}^6 C_3 \times^6 C_4 + {}^6 C_2 \times^6 C_5\]

\[= 2\left( {}^6 C_5 \times^6 C_2 +^6 C_4 \times^6 C_3 \right)\]
\[ = 2\left( 90 + 300 \right)\]
\[ = 2\left( 390 \right)\]
\[ = 780\]