Let $t_n$ denote the number of integral-sided triangles with distinct sides chosen from $\{1, 2, 3, \ldots, n\}$. Then,$t_{20} - t_{19}$ equals

In how many ways can $21$ English books and $19$ Hindi books be placed in a row so that no two Hindi books are together?

Some identical balls are arranged in rows to form an equilateral triangle. The first row consists of one ball,the second row consists of two balls and so on. If $99$ more identical balls are added to the total number of balls used in forming the equilateral triangle,then all these balls can be arranged in a square whose each side contains exactly $2$ balls less than the number of balls each side of the triangle contains. Then the number of balls used to form the equilateral triangle is

The lines $L_1, L_2, \ldots, L_{20}$ are distinct. For $n=1, 2, 3, \ldots, 10$,all the lines $L_{2n-1}$ are parallel to each other,and all the lines $L_{2n}$ pass through a given point $P$. The maximum number of points of intersection of pairs of lines from the set $\{L_1, L_2, \ldots, L_{20}\}$ is equal to:

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

$A$ set of $10$ parallel lines is intersected by another set of $m$ parallel lines. If the total number of parallelograms formed by these two sets of lines is $675$,then $m$ is equal to: