The total number of ways of selecting six coins out of $20$ one rupee coins,$10$ fifty paise coins,and $7$ twenty-five paise coins is:

In a lottery,a person chooses six different natural numbers at random from $1$ to $20$. If these six numbers match the six numbers already fixed by the lottery committee,he wins the prize. What is the probability of winning the prize in the game? [Hint: The order of the numbers is not important.]

If $\binom{n-1}{r} = (k^2 - 3) \binom{n}{r+1}$,then $k \in \dots$

$A$ class contains $b$ boys and $g$ girls. If the number of ways of selecting $3$ boys and $2$ girls from the class is $168$,then $b + 3g$ is equal to.

If ${}^nC_{r-1}=330$,${}^nC_r=462$,and ${}^nC_{r+1}=462$,then $r$ is equal to

If $^nC_3 + ^nC_4 > ^{n+1}C_3$,then