Seven white balls and three black balls are randomly placed in a row. The probability that no two black balls are placed adjacently equals

The number of straight lines that can be formed by joining $8$ points on a circle is:

Let $p_n$ denote the total number of triangles formed by joining the vertices of an $n$-sided regular polygon. If $p_{n+1} - p_n = 66$,then the sum of all distinct prime divisors of $n$ is:

There are $5$ points $P_1, P_2, P_3, P_4, P_5$ on the side $AB$,excluding $A$ and $B$,of a triangle $ABC$. Similarly,there are $6$ points $P_6, P_7, \ldots, P_{11}$ on the side $BC$ and $7$ points $P_{12}, P_{13}, \ldots, P_{18}$ on the side $CA$ of the triangle. The number of triangles that can be formed using the points $P_1, P_2, \ldots, P_{18}$ as vertices is:

There are $n$ straight lines in a plane,no two of which are parallel and no three pass through the same point. Their points of intersection are joined. Then the number of fresh lines thus obtained is

Out of $30$ points in a plane,$8$ of them are collinear. The number of straight lines that can be formed by joining these points is: