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If there are only a handful of objects, then you can count them with a moment's thought, but the techniques of combinatorics can extend to quickly and efficiently tabulating astronomical quantities. See more

If two fair coins are tossed simultaneously, what is the probability that exactly one head appears?

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In a business meeting, a total of 45 handshakes occurred. How many people were at the meeting?

Assume that each person shakes hands with each other person exactly once.

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How many triangles are there in the above image?

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If someone painted the outside of a \( 5 \times 5 \times 5 \) cube made out of \( 1 \times 1 \times 1 \) unit cubes, how many of the unit cubes would have paint on exactly 3 sides?

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