Q:

Number of straight lines can be drawn

How many different straight lines can be drawn using $9$ points on the triangle of the figure below?

My try: Considering points other than $A$, $B$ and $C$ we got $2\times3 + 2\times1 + 3\times1 = 11$. For those three points,  number of lines are $1 \times1 + 2 \times 1 + 3\times1 = 6$. So the total number is $6 + 11 + 3 = 20$ counting sides as possible lines but the solution gives $24$. What are the other lines which I'm not considering here? Also I think there is a combinatorial solution but I didn't find that.

A:

We have $9$ points and a straight line is determined by two distinct points.
We can use the binomial coefficients.

In how many ways can we choose pairs of points?
  $$\binom{9}{2}=\frac{9\cdot 8}{2}=36$$

But, $A,D,E, B$ are collinear as well as $A,F,G,H,C$ and $B,I,C$.
So, we have $\binom{4}{2}+\binom{5}{2}+\binom{3}{2}$ non-distinct lines.
$$\binom{4}{2}+\binom{5}{2}+\binom{3}{2}=\frac{4\cdot 3}{2}+\frac{5\cdot 4}{2}+\frac{3\cdot 2}{2}=19$$
Now, we have to take $AB, BC$ and $AC$ into account, so the final result is:
$$36-19+3=20$$

