Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1} - T_n = 10$,then the value of $n$ is:

There are $10$ points on the line segment $AB$ excluding $A$ and $B$,and $8$ points on the line segment $AC$ excluding $A$ and $C$. The number of triangles formed using these $18$ points (excluding $A, B,$ and $C$) is:

The crew of an $8$-oar boat is to be chosen from $12$ men,of whom $3$ can row on the stroke side only. The number of ways in which the crew can be arranged is

Three parallel straight lines $L_1, L_2$ and $L_3$ lie on the same plane. Consider $5$ points on $L_1, 7$ points on $L_2$ and $9$ points on $L_3$. Then the maximum possible number of triangles formed with vertices at these points is:

$A$ regular polygon has $20$ sides. The number of triangles that can be drawn by using the vertices but not using the sides are

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1, 2, 3, \ldots, 10$,all the lines $L_{2n-1}$ are parallel to each other,and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to: