Question

For an examination, a candidate has to select $7$ subjects from $3$ different groups $A,B,C$ which contain $4,5,6$ subjects, respectively.The number of ways in which a candidate can make his selection if he has to select at least $2$ subjects from each group are

• A. $2700$
• B. $2600$
• C. $2500$
• D. $2400$

Answer 1

Answer: (A)

$2700$

He should select atleast $2$ subjects from each group to make a total of 7 subjects:
So there are three cases:
1. $(A,B,C)\equiv (2,2,3)={}^4{\text{C}}_2\times {}^5{\text{C}}_2\times {}^6{\text{C}}_3=1200$ ways
2. $(A,B,C)\equiv (2,3,2)={}^4{\text{C}}_2\times {}^5{\text{C}}_3\times {}^6{\text{C}}_2=900$ ways
3. $(A,B,C)\equiv (3,2,2)={}^4{\text{C}}_3\times {}^5{\text{C}}_2\times {}^6{\text{C}}_2=600$ ways
So, total $=2700$ ways
Hence, option 'A' is correct.