Find the number of marked points on the plane,if when connected pairwise by line segments,the total number of line segments formed is $15$.

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?

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

On an $n \times n$ chessboard,the total number of rectangles which are not squares is $350$. Then,the number of white squares on the chessboard is .......

If $T_n$ denotes the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides and $T_{n+1}-T_{n}=21$,then $n=$

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