Question

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

Answer 1

Clearly, the candidate may select the questions as under:

(i) (3 out of 7 from A) and (5 out of 5 from B) or

(ii) (4 out of 7 from A) and (4 out of 5 from B) or

(iii) (5 out of 7 from A) and (3 out of 5 from B).

The number of ways of these selections are :

Case I ${}^7{C}_3{\times }^5{C}_5=35\times 1=35.$

Case II ${}^7{C}_4{\times }^5{C}_4=35\times 5=175.$

Case III ${}^7{C}_5{\times }^5{C}_3=21\times 10=210.$

Hence, the required number of ways = $(35+175+210)=420.$