In a row,$6$ boys and $6$ girls are arranged at random. Find the probability that all $6$ girls are together.

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

If $T_n$ denotes the number of triangles which can be formed using the vertices of a regular polygon of $n$ sides and $T_{n+1}-T_{n}=21$,then $n=$

Line $L_1$ with slope $2$ and line $L_2$ with slope $\frac{1}{2}$ intersect at the origin $O$. In the first quadrant,$P_1, P_2, \ldots, P_{12}$ are $12$ points on line $L_1$ and $Q_1, Q_2, \ldots, Q_9$ are $9$ points on line $L_2$. The total number of triangles that can be formed having vertices at three of the $22$ points $(O, P_1, P_2, \ldots, P_{12}, Q_1, Q_2, \ldots, Q_9)$ is:

If the number of diagonals in a polygon with $n$ sides is $275$,then $n = .....$

Out of $20$ points,no three points are collinear except for $4$ points which are collinear. How many lines can be drawn by joining these points?