Let $P_1, P_2, \ldots, P_{15}$ be $15$ points on a circle. The number of distinct triangles formed by points $P_i, P_j, P_k$ such that $i+j+k \neq 15$ is

$A$ regular polygon has $170$ diagonals. Then the measure of the interior angle of the polygon is

The number of triangles whose vertices are at the vertices of a regular octagon but none of whose sides is a side of the octagon is

The number of diagonals in a convex polygon with $n$ sides is .....

Six '$X$'s have to be placed in the squares of the figure such that each row contains at least one '$X$'. In how many different ways can this be done?

There are two women participating with some men in a chess tournament. Each participant played two games with every other participant. The number of games that the men played between themselves is $66$ more than the number of games that the men played with the women. Find the total number of participants in the tournament.