If a committee consists of $3$ men and $2$ women,in how many ways can a committee of $5$ members be formed from $5$ men and $4$ women?

$A$ die is rolled three times. Find the probability of getting a larger number than the previous number each time.

If $^{10}C_r = ^{10}C_{r+2}$,then $^5C_r$ equals

If ${}^n C_r$ denotes the number of combinations of $n$ things taken $r$ at a time,then the expression ${}^n C_{r+1} + {}^n C_{r-1} + 2{}^n C_r$ equals

The number of ways of selecting a committee of $30$ persons from $20$ boys,$20$ girls,and $20$ teachers such that the participation of the number of boys,girls,and teachers in that committee is equal,is

Everybody in a room shakes hands with everybody else. The total number of handshakes is $45$. The total number of persons in the room is