The number of diagonals in a regular polygon of $100$ sides is

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

$6$ points in a plane are joined in all possible ways by indefinite straight lines. If no two lines are coincident or parallel,and no three lines pass through the same point (with the exception of the original $6$ points),the number of distinct points of intersection is equal to:

The number of triangles $(T_n)$ that can be formed using the vertices of a polygon with $n$ sides is given by $T_n = \binom{n}{3}$. If $T_{n+1} - T_n = 21$,then what is the value of $n$?

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:

Out of $n$ points in a plane,$p$ points are collinear. (No three of the remaining points are collinear). The number of lines that can be drawn passing through these points is: