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

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

If there are $10$ coplanar points,out of which $5$ are collinear,how many total triangles can be formed by joining these points?

Some couples participated in a mixed doubles badminton tournament. If the number of matches played,such that no couple played in a match,is $840$,then the total number of persons who participated in the tournament is $........$.

There are $10$ points in a plane of which no three points are collinear except $4$ points. The number of distinct triangles that can be formed by joining these points such that at least one of the vertices of every triangle formed is from the given $4$ collinear points 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