[Math] How many ways he can attempt the paper

Question

Joey has to attempt a question paper that has 3 sections with 6 questions in each section .

If he has to attempt any 8 questions , choosing atleast 2 questions from each section.

Then in how many ways he can attempt the paper?

I tried by taking 2 question in each section as

6C2 ways.

So in all 3 section , he can select 6 questions in 6C2 * 6C2 * 6C2 ways.

Now remaining 2 question he can select from remaining 12 question in 12C2 ways.

So total ways are 6C2 * 6C2 * 6C2 * 12C2 ways.

But the answer is not coming correct.

Is my approach incorrect?

Or i am missing any cases?

Thanks in advance.

Answer 1

First, let's split the problem to subproblems depending on how many questions do we choose from each section.

For example "2 questions from the first section, 4 questions from the second section, 2 questions from the third section" would be one such subproblem. How many are they?

With 8 questions, 3 sections, and at least 2 questions per section, our choices are these: 2+2+4; 2+3+3; 2+4+2; 3+2+3; 3+3+2; 4+2+2. There are essentially just two subproblems, 2+2+4 and 2+3+3, each of them repeated 3 times.

Now, in how many ways can Joey attempt each of these subproblems?

To choose 2+2+4 questions he has 6C2 * 6C2 * 6C4 options. To choose 2+3+3 questions he has 6C2 * 6C3 * 6C3 options. The answer is: 3 * 6C2 * 6C2 * 6C4 + 3 * 6C2 * 6C3 * 6C3.