M&S Combinations

1
Find the number of diagonals that can be drawn in an n-sized polygon
答:
首先﹐因為每一個點可以與另外(n-3)個點形成一條對角線﹐這時對角線總數是n(n-3)。但因為每一條對角線由兩點組成﹐上述計法將同一條對角線算了二次﹐所以實際對角線總數應是n(n-3)/2

2
Given 10 points in a plane, 5 of which lie in the same straight line and no other group of 3 or more points can be drawn through by one straight line. Find the number of different triangles that can be formed by using the 10 points as vertices.
翻譯：一個平面上有10點﹐其中除5點成一直線﹐其餘任意三點皆不能成一直線。問這10點可以組成多少個不同的三角形?
答:
首先因為3點成一三角形﹐這時三角形總數是10C3=120
但由已知直線上5點任取3點的三角形是「直線三角形」要捨去﹐其數目是5C3=10
所以這10點可以組成不同的三角形數目
=10C3-5C3
=120-10
=110