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

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1}-T_n=10$,then the value of $n$ is

In a tournament with five teams,each team plays against every other team exactly once. Each game is won by one of the playing teams and the winning team scores one point,while the losing team scores zero. Which of the following is $NOT$ necessarily true?

If $t_n$ denotes the number of triangles formed with $n$ points in a plane,no three of which are collinear,and if $t_{n+1}-t_n=36$,then $n$ is equal to

$A$ polygon has $44$ diagonals. The number of its sides is

$m$ men and $n$ women are to be seated in a row so that no two women sit together. If $m > n$,then the number of ways in which they can be seated is