Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Here is the question.

Each of the 90 students participated in at least one of the three track events A, B and C. If 20 students participated in A, 40 students participated in B, and 60 students participated in C, and if 5 students participated in all three events, how many students participated in at least two of these events?

Now, I applied the simple inclusion exclusion principle here -

$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \\ |A \cap B| + |A \cap C| + |B \cap C| = |A| + |B| + |C| + |A \cap B \cap C| - |A \cup B \cup C|$

This gives 35 as the result. However the current result is 20 (as per attached explanation). What am I doing wrong here?

enter image description here

share|cite|improve this question
The Venn diagram is there to help. Formulas tend to look all alike: it is good to look at the picture and see what is counted by $|A\cap B|+|A\cap C|+|B\cap C|$. – André Nicolas Sep 1 '12 at 0:10
up vote 1 down vote accepted

The problem is that $|A \cap B| + |A \cap C| + |B \cap C|$ isn’t the number who participated in at least two events: every student who participated in all three events got counted in $|A\cap B|$, again in $|A\cap C|$, and yet a third time in $|B\cap C|$. Those five students have therefore been counted three times each. In the notation of the attached explanation, $$|A \cap B| + |A \cap C| + |B \cap C|=(b+e)+(d+e)+(f+e)=b+d+f+3e\;.$$

If you want the number who participated in at least two events, you want to count each of those five people once, not three times, so the correct answer is $35-2\cdot5=25$. If you want the number who participated in exactly two events, you don’t want to count those five people at all, so the correct answer is $35-3\cdot5=20$. The attached explanation gives the answer to this second question, not to the one that you asked at the beginning of the post.

share|cite|improve this answer
Cool. Thanks a lot! – user957 Aug 31 '12 at 23:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.