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?

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

In how many ways can $5$ boys and $5$ girls stand in a row so that no two girls may be together?

There are $n$ distinct points on the circumference of a circle. If the number of pentagons that can be formed using these points as vertices is equal to the number of triangles that can be formed,then what is the value of $n$?

If three vertices are chosen from the six vertices of a regular hexagon and joined to form a triangle,what is the probability that the triangle formed is an equilateral triangle?

The total number of possible ways in which a person can go from $A(0,0)$ to $B(3,3)$ if he can travel only to the right or upward along the grid lines is: