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How many unique quadrilaterals can be inscribed in the verti

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Manager
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How many unique quadrilaterals can be inscribed in the verti [#permalink] New post 12 Nov 2017, 01:18
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How many unique quadrilaterals can be inscribed in the vertices of a nonagon (a 9-sided figure), if points A and B, two vertices in the nonagon, cannot make up the same quadrilateral?

(A) 126

(B) 105

(C) 96

(D) 65

(E) 21

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Director
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Re: How many unique quadrilaterals can be inscribed in the verti [#permalink] New post 13 Nov 2017, 00:24
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The total number of quadrilaterals that can be inscribed in a nonagon can be computed by counting the number of ways 4 points can be chosen out of 9, i.e. 9C4 = \frac{9!}{5!4!} = 126.

But this is not what we are looking for. Indeed, we need to exclude those quadrilaterals which have two points being A and B. To compute them, given that two points are fixed, we can count how many ways are there to choose 2 points out of 7, i.e. 7C2 = \frac{7!}{2!5!} = 21. Subtracting 21 from 126, we get 105.

Answer B
Re: How many unique quadrilaterals can be inscribed in the verti   [#permalink] 13 Nov 2017, 00:24
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How many unique quadrilaterals can be inscribed in the verti

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