Combinations

[Dean54321]

Twelve points are arranged in order around a circle
a) How many triangles can be drawn with these points as verticies?
b) How many pairs of such triangles can be drawn if the vertices of the two triangles are distinct?
c) In how many such pairs will the triangles i) Not overlap ii) Overlap

For part a, I just did 12C3

For part b, I know you times 12C3 by 9C3, but I'm confused when and why you divide by 2.

Finally, I have no idea how to do the last question. I tried solving for 2 distinct points and one other point.

 

[pka]

Twelve points are arranged in order around a circle
a) How many triangles can be drawn with these points as verticies?
b) How many pairs of such triangles can be drawn if the vertices of the two triangles are distinct?
c) In how many such pairs will the triangles i) Not overlap ii) Overlap
For part a, I just did \(\dbinom{12}{3}\)

Correct1
For part b) we need to divide the twelve points into four unordered partitions.
OR form four unnamed sets of three each: \(\large\dfrac{12!}{(3!)^4(4!)}\)

 

[Dr.Peterson]

Twelve points are arranged in order around a circle
a) How many triangles can be drawn with these points as vertices?
b) How many pairs of such triangles can be drawn if the vertices of the two triangles are distinct?
c) In how many such pairs will the triangles i) Not overlap ii) Overlap

For part a, I just did 12C3

For part b, I know you times 12C3 by 9C3, but I'm confused when and why you divide by 2.

Finally, I have no idea how to do the last question. I tried solving for 2 distinct points and one other point.

(a) is good.

(b) is correct; you have to divide by 2 because you can choose the same two triangles in two orders, so you've overcounted by a factor of 2.

(c) I would think about how many different pairs of triangles involve any given set of 6 points, and how many of those do not overlap.