There are $3, 4,$ and $5$ points on the sides $AB, BC,$ and $CA$ of a triangle $ABC$ respectively. How many triangles can be formed using these points as vertices?

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1, 2, 3, \ldots, 10$,all the lines $L_{2n-1}$ are parallel to each other,and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to:

How many triangles $(T_n)$ can be formed by joining the vertices of a regular polygon with $n$ sides? If $T_{(n+1)} - T_n = 10$,what is the value of $n$?

There are $12$ points in a plane,no three of which are in the same straight line,except $5$ points which are collinear. Then the total number of triangles that can be formed with the vertices at any three of these $12$ points is

Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

In how many ways can $5$ boys and $3$ girls sit in a row so that no two girls are together?