An examination paper consists of $12$ questions divided into two parts $A$ & $B$ . Part $A$ , containing $7$ questions & part $B$ contains $5$ questions. A candidates is required to attempt $8$ questions selecting at least $3$ from each part. In how many ways can the candidates select the questions?

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Solution

Student has to do $8$ questions such that he does at least $3$ questions from both the sets
How can he go about (Set I, Set II) $= (3, 5)$ or $(4, 4)$ or $(5, 3)$
So, total number of ways of doing this is given by
$5 C_{3} \times 7 C_{5}$ + $5 C_{4} \times 7 C_{4}$ + $5 C_{5} \times 7 C_{3}$

$= (10) (21) + (5) (35) + (1) (35) = 210 + 210 = 420$

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