# Thread: Inclusion-exclusion or difference rule for this problem?

1. ## Inclusion-exclusion or difference rule for this problem?

Hi guys, I have a problem, and I'm not sure which method I should be using. I didn't want to get too far into the work before I realise that I've chosen the wrong method.

The problem is:

in a class, three tests were given. out of ... students in the class:

... did well on test A
... did well on test B
... did well on test C

... did well on tests A and B
... did well on tests A and C
... did well on tests B and C

... did well on all three tests

How many did not do well on any test?
I'm pretty sure I should be using inclusion-exclusion for the problem, but as I'm not very good with mathematics, I'm not sure. Any help you guys could provide would be great.

Edit: After playing around with this, I've tried using the difference rule and I've gotten an answer that's correct to me.

Out of ten students (labelled 1-10):

4 did well on test A (1, 2, 3, 6)
4 did well on test B (2, 3, 6, 8)
3 did well on test C (6, 8, 9)

Using those values, just by looking at it I can see that students 4, 5, 7 and 10 didn't do well on any test.

Let T = all the students, let A be students who did well on test A, B be students who did well on B, and C be students who did well on test C, that you can get the answer by doing | T | - | A U B U C | (the difference rule) which gives four, which is the correct answer from above.

So I think I'm doing it correctly, but I can't understand the relevance of all other data? e.g. students who did well on A and B, B and C, or on all three tests? Is there something I'm missing, or are they unnecessary for answering this question?

2. If the students are labelled, and you are told the labels of the students who did well in test A, and the labels of those who did well in test B and likewise for test C, then your method of solving the problem is all good.

If you didn't know which student had done well on which test among A, B and C, then using the inclusion-exclusion principle would be the way to go.

3. Awesome! Thanks heaps for that HappyJoe. That was exactly the thing I was curious about.

And for the record, The inclusion-exclusion principle was what I used, for the exact reason you said.