# How many pentagons are there such that all their vertices are points from this set of twelve points and each of the pentagons has at least one vertex of each colour?Twelve points are placed on the...

How many pentagons are there such that all their vertices are points from this set of twelve points and each of the pentagons has at least one vertex of each colour?

Twelve points are placed on the circumference of a circle. Five of these twelve points are coloured black and the other seven green. How many pentagons are there such that all their vertices are points from this set of twelve points and each of the pentagons has at least one vertex of each colour?

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### 1 Answer

There are 12 points -- 5 colored black and 7 colored green. We want the number of pentagons such that there is at least 1 vertex of each color.

There are `_12C_5=792` possible pentagons. (The number of combinations of 12 taken 5 at a time. Order does not matter.)

The number of pentagons containing only black vertices is `_5C_5=1`

The number of pentagons containing only green vertices is `_7C_5=21`

Thus of the 792 possible pentagons, 22 are unicolored.

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**There are 792-22=770 different pentagons with at least 1 of each color vertices.**

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