Suppose $BC$ is a given line segment in the plane and $T$ is a scalene triangle. The number of points $A$ in the plane such that the triangle with vertices $A, B, C$ (in that order) is similar to triangle $T$ is

$A$ simple graph contains $24$ edges. The degree of each vertex is $3$. The number of vertices is:

How many triangles can be formed by joining four points on a circle?

In a plane,there are $37$ straight lines,of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides,no three lines pass through one point,no line passes through both points $A$ and $B$,and no two lines are parallel. The number of intersection points the lines have is equal to:

In how many ways can $5$ boys and $5$ girls be arranged in a row such that no two girls are together?

There are $3$ girls in a class of $10$ students. The number of different ways in which they can be seated in a row such that no two of the three girls are together is