ine and Angel are twin sisters in the second grade who share a room and have a pet mouse collection of 8 mice. Angel says she adores 6 of them, but Jane says $$ t $$ at is impossible, since she loves 7 and there are not 13 mice. Is Jane right? How would you explain to the sisters the conceptual reason for your answer? A. No, Jane is not right. B. Yes, Jane is right.

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1. We are given that there are a total of $$8$$ mice.
2. Angel claims she adores $$6$$ of them and Jane claims that she loves $$7$$ of them.
3. At first glance, one might add the two numbers:  
   $$
   6 + 7 = 13.
   $$
   However, this sum would only be valid if there were no overlap in the mice they both adore.
4. In reality, some mice can be adored by both Angel and Jane. Let the number of mice that both sisters adore be $$x$$.
5. Then, the total number of mice can be expressed as:  
   $$
   \text{Total mice} = 6 + 7 - x = 13 - x.
   $$
6. Since we know there are only $$8$$ mice, we set up the equation:  
   $$
   13 - x = 8.
   $$
7. Solving for $$x$$, we get:  
   $$
   x = 13 - 8 = 5.
   $$
8. This means that $$5$$ mice are adored by both sisters. Therefore, Angel adores $$6$$ mice (with $$5$$ in common) and Jane adores $$7$$ mice (also with $$5$$ in common), which is perfectly possible with only $$8$$ mice.
9. The conceptual mistake in Jane's reasoning is adding the individual numbers without considering the overlap; when two sets have common elements, the total number of distinct elements is not the simple sum of the two counts.

The correct answer is:  
A. No, Jane is not right.
No, Jane is not right. When both sisters adore some of the same mice, simply adding their individual counts (6 + 7) would count those shared mice twice. Since there are only 8 mice in total, the overlap means that Jane's total of 7 adored mice is possible without exceeding the total number of mice.

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