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?

There are $16$ points in a plane out of which $6$ are collinear. How many lines can be drawn by joining these points?

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle,then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

The straight lines $l_1, l_2, l_3$ are parallel and lie in the same plane. $A$ total number of $m$ points are taken on $l_1$,$n$ points on $l_2$,and $k$ points on $l_3$. The maximum number of triangles formed with vertices at these points is:

Ten points lie in a plane such that no three of them are collinear. The number of lines passing through exactly two of these points and dividing the plane into two regions,each containing four of the remaining points,is

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