MathCounts 1993 National Sprint #28

Question

Here is the problem: In a small town of 100 men, 85 are married, 70 have a telephone, 75 own a car, and 80 own their own home. On this basis, what is the smallest possible number of men who are married, have their own telephone, own their own car, and own their own home?

I drew the following diagram:

I then found the intersection of all the sets (own a car, own a house, have a phone, are married), which is shaded with a darker tone of blue. This intersection contains 10 squares. In other words, this intersection represents $10 \cdot 5= \boxed{50}$ men. My answer is incorrect. The correct answer is $\boxed{10}$ men.

Can you explain why my solution doesn't work? (I would prefer if you gave me an algebraic or pictorial explanation, but if this is not possible then it's okay.)

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Anyway, someone else provided a solution which yielded the correct answer. Can you explain why his/her solution works? (Once again, I would prefer if you gave me an algebraic or pictorial explanation, but if this is not possible then it's okay.)

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Final question: How does inclusion-exclusion work in this problem? (I would prefer if you gave me a pictorial or algebraic explanation. If this is not possible then it's okay.)

Answer 1

70 people have a telephone. That means that 30 people do NOT have a telephone. 20 men don't own their home, 15 men aren't married, and 25 men don't own a car.

The stars below represent 5 men per star. Now the dashes below those stars can be seen as kind of eliminating those stars for each category. The point of this question is to minimize the number of stars that are not overlapped. One way of doing this is below:

* * * * * * * * * * * * * * * * * * * *
- - - - - - (no telephone)
- - - - (no home)
- - - (no chick)
- - - - (no car)

This distribution allows 30 people to not have atleast one thing, which means that 100-30=70 people will have all four. So how do we minimize the number of people that have all four? Let's try to minimize the amount of overlap for the categories of not having something. So something like this

* * * * * * * * * * * * * * * * * * * *
- - - - - - (no telephone)
            - - - - (no home)
                    - - - (no chick)
                          - - - - - (no car)

90 men don't have at least one thing, which means that 100-90=10 people have all four things.

Inclusion exclusion questions involve finding the amount of things that are "included" in some category by finding the people that are excluded from that category. 90 men were excluded from the category of not having all four things, and so we are left with 10 people who have all four.