Question

A question paper consists of $12$ questions divided into two parts $A$ and $B$. $A$ contains $7$ questions and $B$ contains $5$ questions. In how many ways can a student attempt the question papers such that at least $3$ should be attempted from each part

• A. $420 $
  
• B. $280 $
• C. $360 $
• D. $460 $

Answer 1

The correct option is A $420 $

Ways of selecting 8 questions

Case 1:

3 questions out of 5

5 questions out of 7

Number of ways of selecting ${=}^5{C}_3{\times }^7{C}_5$

$=\frac{5!}{3!\times 2!}\times \frac{7!}{5!\times 2!}$

$=5\times 2\times 7\times 3$

$=210$

Case 2:

4 questions out of 5

4 questions out of 7

Number of ways of selecting ${=}^5{C}_4{\times }^7{C}_4$

$=\frac{5!}{4!\times 1!}\times \frac{7!}{4!\times 3!}$

$=5\times 7\times 5$

$=175$

Case 3:

5 questions out of 5

3 questions out of 7

Number of ways of selecting ${=}^5{C}_5{\times }^7{C}_3$

$=\frac{5!}{5!\times 0!}\times \frac{7!}{3!\times 4!}$

$=1\times 7\times 5$

$=35$

Total ways $=210+175+35=$ 420 ways