Inclusion/Exclusion Counting

Question

At a particular school, all students participate in at least one out of three activities: debate, music, and community service. 3 students do all three activities, 21 students do two or more activities, half of all students participate in music, and the ratio of participants in debate, music, and community service, respectively, is $5:4:3$. What is the total number of students at the school?

So I gave Debate 5x people, music 4x people, and community service 3x people. So the total number of people is $5x+4x+3x-21+3= 12x-18$. Now music is half of the total, or $6x-9$. But this is equal to $4x$, so we have $6x-9=4x$. But then x is not a whole number! Where did I go wrong?

Answer 1

The number of students involved in two or more clubs is

$$21=|(A\cap B)\cup (A\cap C)\cup (B\cap C)|$$

Applying inclusion-exclusion to this gives us...

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

$~$

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

And so $|A\cap B|+|A\cap C|+|B\cap C|$ is equal to

$21+6=27$

Using this new corrected value in the inclusion-exclusion expansion in your original attempt:

$5x+4x+3x-27+3=12x-24$

And so...