Question
A question paper on mathematics contains $12$ questions divided into three parts A, B and C, each containing $4$ questions. In how many ways can an examinee answer $5$ questions, selecting at least one from each part.
Attempt
Firstly, I selected three questions (one from each part) and it can be done $4 \cdot 4 \cdot 4$ ways. And hence the remaining two positions for two questions can be given in $^9{\mathrm C}_2$ since there is no restrictions now. So, total ways is $36 \times 64=2304$.
But, in the answer given in the solution manual is $624$. And the process described is:
A B C
1 1 3
1 2 2
which is then arranged for part to be $3 \times (4 \times 4 \times 4 + 4 \times 6 \times 6) = 624$.
Why my process is incorrect? I understand the second solution but, unable to find any fault in my attempt. Please explain.
Best Answer
Let's compare your method with the correct solution.
Solution: As you noted, either three questions are selected from one section with one each from the other two sections or two questions each are selected from two sections and one is selected from the remaining section.
Three questions from one section and one question each from the other two sections: Select the section from which three questions are drawn, select three of the four questions from that section, and select one of the four questions from each of the other two sections. This can be done in $$\binom{3}{1}\binom{4}{3}\binom{4}{1}\binom{4}{1}$$ ways.
Two questions each from two sections and one question from the remaining section: Select the two sections from which two questions are drawn, select two of the four questions from each of those sections, and select one of the four questions from the other section. This can be done in $$\binom{3}{2}\binom{4}{2}\binom{4}{2}\binom{4}{1}$$ ways.
Total: Since the two cases are mutually exclusive and exhaustive, there are $$\binom{3}{1}\binom{4}{3}\binom{4}{1}\binom{4}{1} + \binom{3}{2}\binom{4}{2}\binom{4}{2}\binom{4}{1} = 624$$ ways to select five questions so that at least one is drawn from each of the three sections.
Why your method is wrong?
You are counting each selection in which three questions are drawn from one section and one question is drawn from each of the other sections three times, once for each way you could designate one of the three questions as the question that is drawn from that section. For example, suppose questions $A_1, A_2, A_3, B_1, C_1$ are selected. You count this selection three times.
\begin{array}{c c} \text{designated questions} & \text{additional questions}\\ \hline A_1, B_1, C_1 & A_2, A_3\\ A_2, B_1, C_1 & A_1, A_3\\ A_3, B_1, C_1 & A_2, A_3 \end{array}
You are counting each selection in which two questions each are drawn from two sections and one question is drawn from the other section four times, two times for each way you could designate one of the two question from each section from which two questions are drawn as the question that is drawn from that section. For instance, if questions $A_1, A_2, B_1, B_2, C_1$ are drawn, your method counts this selection four times.
\begin{array}{c c} \text{designated questions} & \text{additional questions}\\ \hline A_1, B_1, C_1 & A_2, B_2\\ A_1, B_2, C_1 & A_2, B_1\\ A_2, B_1, C_1 & A_1, B_2\\ A_2, B_2, C_1 & A_1, B_1 \end{array}
Notice that $$\binom{3}{1}\color{red}{\binom{3}{1}}\binom{4}{3}\binom{4}{1}\binom{4}{1} + \binom{3}{2}\color{red}{\binom{2}{1}}\binom{4}{2}\color{red}{\binom{2}{1}}\binom{4}{2}\binom{4}{1} = 2304$$