<Question>
In a class of 50 students, 28 participate in MATHCOUNTS, 21 participate in science club, and 6 students participate in neither. How many students participate in both MATHCOUNTS and science club?
</Question>

<think>
<Plan_1>
Okay, let's try to figure out how many students are in both MATHCOUNTS and the science club. Hmm, so we have a class of 50 students. Out of these, 28 are in MATHCOUNTS, 21 are in the science club, and 6 aren't in either. I need to find the overlap between the two groups, right?

Wait, this sounds like a problem where I can use the principle of inclusion-exclusion. Let me recall how that works. The formula is: the number of students in either MATHCOUNTS or the science club (or both) is equal to the number in MATHCOUNTS plus the number in the science club minus the number in both. So, |M ∪ S| = |M| + |S| - |M ∩ S|.
But first, how many students are in either MATHCOUNTS or the science club? Since 6 students are in neither, that means 50 - 6 = 44 students are in at least one of the two clubs. So, |M ∪ S| is 44.

Plugging into the formula: 44 = 28 + 21 - |M ∩ S|. Let me check that.
28 plus 21 is 49. So, 49 minus the number in both equals 44. Therefore, |M ∩ S| = 49 - 44 = 5.

Wait, so does that mean 5 students are in both clubs? Let me verify to make sure I didn't make a mistake.
Let's see. Total students in MATHCOUNTS only would be 28 - 5 = 23. Science club only is 21 - 5 = 16. Then, 23 (M only) + 16 (S only) + 5 (both) + 6 (neither) equals 23 + 16 is 39, plus 5 is 44, plus 6 is 50. That adds up correctly.

But wait, another way to think about it: the total number in both clubs combined, if there was no overlap, would be 28 + 21 = 49. But since there are only 44 students who are in at least one club, the overlap is 49 - 44 = 5.
Yep, that makes sense. So the answer should be 5 students.

I don't think I made any mistakes here. The key was remembering to subtract the number of students not in either club to find out how many are in at least one, then apply inclusion-exclusion.
The check by adding up all the parts also helps confirm that 5 is the right answer.

**Final Answer**
The number of students participating in both MATHCOUNTS and the science club is \boxed{5}.
</Plan_1>

<Execution_1>
No prompt to execute.
</Execution_1>

<Final_answer>
Therefore, the number of students who participate in both MATHCOUNTS and the science club is $\boxed{5}$.
</Final_answer>
</think>