Number of integral solutions to the equation $x+y+z=21$, where $x \geq 1, y \geq 3, z \geq 4$, is equal to $……….$.

A man has $7$ friends. In how many ways he can invite one or more of them for a tea party

Determine $n$ if

$^{2 n} C_{3}:\,^{n} C_{3}=12: 1$

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

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