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 of $n$ sides has $170$ diagonals,then $n$ is equal to

The number of arrangements of the letters of the word $PALANHAR$ in which no two vowels are together and exactly two vowels are at odd places,is

Consider a rectangle $ABCD$ having $5, 6, 7, 9$ points in the interior of the line segments $AB, BC, CD, DA$ respectively. Let $\alpha$ be the number of triangles having these points from different sides as vertices and $\beta$ be the number of quadrilaterals having these points from different sides as vertices. Then $(\beta-\alpha)$ is equal to:

The greatest possible number of points of intersection of $8$ straight lines and $4$ circles is

$A$ set of $10$ parallel lines is intersected by another set of $m$ parallel lines. If the total number of parallelograms formed by these two sets of lines is $675$,then $m$ is equal to: