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

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

$A$ rectangle with side lengths $2m - 1$ and $2n - 1$ is divided into unit squares by drawing parallel lines as shown. How many rectangles are there such that both side lengths are odd?

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?

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

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