There are points of A~F which equally divide the circumference of the circle with a center point O into six parts as shown in the figure below.
Make a triangle by connecting three points out of seven (O and A~F).
How many triangles can you make?
Make a triangle by connecting three points out of seven (O and A~F).
How many triangles can you make?
Answer
In case of counting the number of the triangles which connect point O, since two other than the point O will be chosen from six points of A~F, it is (6 × 5) / (2 × 1) =15 pieces.
However, since AOD, BOE, and COF do not become a triangle, they are not counted. then it is 15 - 3 = 12 pieces.
Therefore it is 20 + 12 = 32 pieces in total.
32 pieces
Solution
In case of counting the number of the triangles which do not connect point O, since three points will be chosen from six points of A~F, it is (6 × 5 × 4) / (3 × 2 × 1) =20 pieces.
In case of counting the number of the triangles which connect point O, since two other than the point O will be chosen from six points of A~F, it is (6 × 5) / (2 × 1) =15 pieces.
However, since AOD, BOE, and COF do not become a triangle, they are not counted. then it is 15 - 3 = 12 pieces.
Therefore it is 20 + 12 = 32 pieces in total.
