If $t_n$ denotes the number of triangles formed with $n$ points in a plane,no three of which are collinear,and if $t_{n+1}-t_n=36$,then $n$ 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

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?

Three of the six vertices of a regular hexagon are chosen at random. The probability that the triangle formed by these three vertices is equilateral is equal to

The number of ways in which the letters $A, B, C, D, E$ can be placed in the $8$ boxes of the figure below so that no row remains empty and at most one letter can be placed in a box is:

The number of straight lines that can be formed by joining $20$ points,no three of which are in the same straight line,except $4$ of them which are in the same line,is: