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

The number of ways in which $3$ identical balls can be distributed into $7$ distinct bins is

$^{n-1}C_r = (k^2 - 8) ^nC_{r+1}$ if and only if:

If $10 \cdot ^nC_2 = 3 \cdot ^{n+1}C_3$,then the value of $n$ is

$A$ company has $10$ employees. The company has decided to form a team including at least three employees and also excluding at least three employees. Then the number of ways of forming the team is

In how many ways can two balls of the same color be selected from $4$ distinct black balls and $3$ distinct white balls?