How many triangles can be drawn with the vertices of a regular hexagon. Please give me explanation for your answer. Thanks in advance?

Hello my dear selfee, let us mark the vertices by numbers 1,2,3,4,5 and 6

Now let us list out the possibilities

123, 124, 125, 126, 234, 235, 236, 341, 345, 346, 451, 452, 456, 561, 562, 563, 613, 614. So far total has come to 18

Now we can also have 135, 246. Hence the total will be 20 triangles. Hope this would be the right and final answer.

So it establishes that having three boxes and six digits 1,2,3,4,5 and 6, the boxed can be filled without repetition in 6C3 ways. So 6 x 5 x 4 / 1 x 2 x 3 = 20.

18

6 internes

12 externs

6 take 2 gives you two vertices, then 4 take 1 gives you the third vertex so

C(6,2) * C(4,1) =15 * 4 = 60.

Alternative solution: 6 take 1 for the first vertex, then 5 take 2 for the remaining two vertices, so

C(6,1) * C(5,2) = 6 * 10 = 60.