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1.直接求解例1从平面α上取6点,从平面β上取4点,这10个点最多可以确定多少个三棱锥?“和”的思想要想使这10个点构成的三棱锥最多,除α上6点共面,β上4点共面外,应再无四点共面及三点共线.所以可从平面α上6个点中任取一个与平面β上4个点中任取3个构成三棱锥,有C_6~1C_4~3个;也可以从平面α上6个点中任取2个与平面β上4个点中任取2个构成三棱锥,有C_6~2C_4~2个;还可从平面α上6个点中任取3个与平面β上4个点中任取1个构成三棱锥,有C_6~3C_4~1个.
1. Direct Solution Example 1 Take 6 points from the plane α and 4 points from the plane β. How many triangular pyramids can be determined by the 10 points? The idea of the “sum” is to make the triangle pyramid of the 10 points. At most, except that 6 points on α are coplanar and 4 points on 4 are coplanar, there should be no 4 points coplanar and 3 points collinear. So you can take any one of the 6 points on the plane α and 4 on the plane β. In the point, any three triangular pyramids are included, and there are C_6~1C_4~3; any two of the six points on the plane α and two of the four points on the plane β may constitute three pyramids, and there are C_6 ~2C_4~2; It is also possible to take 3 out of 6 points on the plane α and take 1 out of 4 points on the plane β to form a triangular pyramid with C_6~3C_4~1.