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How do I use Combinatorics to read Venn diagrams?

The problem looks like this:

Stephen asked 100 coffee drinkers whether they like cream or sugar in their coffee. According to the Venn diagram below, how many like:

a) Cream?

b) Sugar?

c) Sugar but not cream?

d) Cream but not sugar?

e) Cream and sugar?

f) Cream or sugar?

g) Black (no cream, no sugar)?

Also, the Venn Diagram picture is attached. In the "Cream" section of the Venn diagram, it states that there are 16 coffee drinkers who like cream. There are 35 for sugar, and in the overlap area there are 20.

How do I solve these problems? Am I over-thinking it?

Re: How do I use Combinatorics to read Venn diagrams?

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Originally Posted by

**alexthesauceboss** How do I solve these problems? Am I over-thinking it?

Possibly. For a start, I assume you can answer c), d) and e). The answers come straight from the picture; no arithmetic is necessary.

Re: How do I use Combinatorics to read Venn diagrams?

Quote:

Originally Posted by

**emakarov** Possibly. For a start, I assume you can answer c), d) and e). The answers come straight from the picture; no arithmetic is necessary.

c) 55 (sugar but not cream I assume means the overlap and the sugar section added together)

d) 36 (cream but not sugar I assume means the overlap and the cream section added together)

e)20 (I assume cream or sugar means just the overlap)

I'll be back in about 3 and a half hours if someone wants to continue to help me.

Re: How do I use Combinatorics to read Venn diagrams?

Quote:

Originally Posted by

**alexthesauceboss** c) 55 (sugar but not cream I assume means the overlap and the sugar section added together)

This cannot be right for two reasons. First, I said that no arithmetic is necessary. Second, the overlap represents those who like *both* cream and sugar; in particular, *they like cream*. And here you are asked about those who like sugar *but not cream*!.

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Originally Posted by

**alexthesauceboss** d) 36 (cream but not sugar I assume means the overlap and the cream section added together)

Wrong for the same reasons.

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Originally Posted by

**alexthesauceboss** e)20 (I assume cream or sugar means just the overlap)

Correct.

Re: How do I use Combinatorics to read Venn diagrams?

Quote:

Originally Posted by

**emakarov** This cannot be right for two reasons. First, I said that no arithmetic is necessary. Second, the overlap represents those who like *both* cream and sugar; in particular, *they like cream*. And here you are asked about those who like sugar *but not cream*!.

Wrong for the same reasons.

Correct.

So then C) is 35, D) is 16, and e) is 20...Do the rest require combinatorics?

Re: How do I use Combinatorics to read Venn diagrams?

Quote:

Originally Posted by

**alexthesauceboss** So then C) is 35, D) is 16, and e) is 20...

Yes.

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Originally Posted by

**alexthesauceboss** Do the rest require combinatorics?

Your answers in post #3 are in fact answers to b) and a). (Why?) For f), note that the three areas in the picture are disjoint, i.e., no person belongs to two groups simultaneously. This means that to find the number of people in all three groups you just need to add the numbers. Finally, that sum is going to be less than 100, and the rest like their coffee black.