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There are nine points on a piece of paper. No three of the points are collinear. How many different triangles can be formed by using three of the nine points as vertices?

  • math -

    Number of possible choices for the first point = 9.

    Number of possible choices for the second point = 8.

    Number of possible choices for the third point = 7.

    Possible triangles with specific order of points = 9*8*7 = 504.

    However, when we say triangle, we are not really concerned in which order the points are selected. So we have over-counted the number of triangles by 6, which is the number of ways to order three points.

    The number of distinct triangles is therefore 504/6=84.

    This number is mathematically called
    9 choose 3, calculated by
    9!/(3!(9-3)!) = 84
    where 9! is factorial 9, = 9*8*7*...*2*1

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