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?

Consider a rectangle $ABCD$ having $5, 6, 7, 9$ points in the interior of the line segments $AB, BC, CD, DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta-\alpha)$ is equal to:

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 $21$ white balls and $19$ black balls be arranged in a row such that no two black balls are together?

$15$ lines are concurrent at a point $P$. $A$ line $L$ not passing through $P$ intersects all the $15$ lines and forms triangles with them. Then the number of triangles having $L$ as one of its sides is

Let $X = \{(a, b) \mid a, b \in \mathbb{Z}, 0 \leq a \leq 20, 0 \leq b \leq 15\}$. Then the number of rectangles with vertices in set $X$ is