How many triangles can you draw using the dots in the picture as vertices?

Question

How many triangles can you draw using the dots below as vertices?

(a) Find an expression for the answer which is the sum of three terms involving binomial coefficient.

(b) Find an expression for the answer which is the difference of two binomial coefficient.

(c) Generalize the above to state and prove a binomial identity using a combinatorial proof. Say you have $x$ points on the horizontal axis and $y$ points in the semi-circle.

Please can someone help me in these kind of sums!!

My work

Finally I got it first from five semicircle points select any 3 points to be vertices of triangle =10 ways second from 7 horizontal points select any 2 points and from 5 semicircle points select any 1 point = 21*5 = 105 ways Third from 7 horizontal points select any 1 points and from 5 semicircle points select any 2 point = 70 ways Thus 70 + 105 + 10=185 triangles are possible.

Answer 1

finally I got it first from five semicircle points select any 3 points to be vertices of triangle =10 ways second from 7 horizontal points select any 2 points and from 5 semicircle points select any 1 point = 21*5 = 105 ways Third from 7 horizontal points select any 1 points and from 5 semicircle points select any 2 point = 70 ways Thus 70 + 105 + 10=185 triangles are possible.

Answer 2

Hint for (a). Any proper triangle will have $0$, $1$, or $2$ points along the horizontal line.

Hint for (b). A proper triangle will have the $3$ vertices not all along the same line. Now we have $12$ points of which $7$ are along the horizontal line.

Now it's your turn!!