# There are 3 sections in a question paper with 5 questions each.

There are 3 sections in a question paper each containing 5 questions. A candidate has to solve only 5 questions, choosing at least one question from each section. In how many ways can he make his choice?

I have thought of a solution but I am over counting the number of ways.

No. of ways to choose one question from each section = (5C1)^3 No. of questions remaining = 12 No. of ways to pick 2 questions from remaining 12 questions = 12 * 11 Total number of ways = (5C1)^3 * 12 * 11

Can somebody tell me where I'm going wrong

• What is the answer? – callculus Oct 19 '18 at 16:18

He can choose the following combinations of questions from each section:

$$(2,2,1);(2,1,2);(1,2,2);(3,1,1);(1,3,1);(1,1,3)$$

Each of the first three combinations has $$\binom{5}{2}\cdot \binom{5}{2}\cdot \binom{5}{1}$$ ways. And each of the next three combinations has $$\binom{5}{3}\cdot \binom{5}{1}\cdot \binom{5}{1}$$ ways.

In total he can make $$3\cdot \left(\binom{5}{2}\cdot \binom{5}{2}\cdot \binom{5}{1}+ \binom{5}{3}\cdot \binom{5}{1}\cdot \binom{5}{1}\right)=2250$$ choices.

For every section we have one path. We choose 3 question from each section. For every path we have to add ($$\text{not multiply}$$) the ways

$$\left( \binom{5}{1}+\binom{5}{1}+\binom{5}{1}\right)$$

Now we make a case decision.

a) We choose 2 questions from one (other) section and 2 questions from the remaining section.

$$\binom{5}{2}\cdot \binom{5}{2}$$

b) We choose 1 questions from one (other) section and 3 questions from the remaining section.

$$\binom{5}{1}\cdot \binom{5}{3}$$

Therefore in total we have

$$\left( \binom{5}{1}+\binom{5}{1}+\binom{5}{1}\right)\cdot \left(\binom{5}{2}\cdot \binom{5}{2}+ \binom{5}{1}\cdot \binom{5}{3}\right)=2250$$

• can you tell me where I'm going wrong? – Rishabh Khandelwal Oct 19 '18 at 16:22
• One problem is $\binom{5}{1}^3$. It must be $\binom{5}{1}^1+\binom{5}{1}^1+ \binom{5}{1}^1$ – callculus Oct 19 '18 at 16:44
• Still answer won't match – Rishabh Khandelwal Oct 19 '18 at 16:55
• What is the answer? – callculus Oct 19 '18 at 16:58
• 2250 is the correct answer – Rishabh Khandelwal Oct 19 '18 at 17:00