In how many ways can $5$ boys and $5$ girls stand in a row so that no two girls may be together?

Let $t_n$ denote the number of integral-sided triangles with distinct sides chosen from $\{1, 2, 3, \ldots, n\}$. Then,$t_{20} - t_{19}$ equals

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

The number of diagonals of a polygon is $35$. If $A$ and $B$ are two distinct vertices of this polygon,then the number of all those triangles formed by joining three vertices of the polygon having $AB$ as one of its sides is:

$A$ polygon has $35$ diagonals,then the number of its sides is

In a plane,there are $10$ points,out of which $4$ are collinear. The number of triangles that can be formed by joining these points is: