Five boys and three girls are seated in a row at random. What is the probability that all three girls sit together?

$A$ parallelogram is cut by two sets of $m$ lines parallel to its sides. The number of parallelograms thus formed is

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?

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 of a regular polygon is $35$,then the number of sides of the polygon is

$T_m$ denotes the number of triangles that can be formed with the vertices of a regular polygon of $m$ sides. If $T_{m+1}-T_m=15$,then $m$ is equal to