In a club election the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote be $62,$ then the number of candidates is :-

There are two urns. Urm $A$ has $3$ distinct red balls and urn $B$ has $9$ distinct blue balls. From each urm two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

Let $A_1,A_2,........A_{11}$ are players in a team with their T-shirts numbered $1,2,.....11$. Hundred gold coins were won by the team in the final match of the series. These coins is to be distributed among the players such that each player gets atleast one coin more than the number on his T-shirt but captain and vice captain get atleast $5$ and $3$ coins respectively more than the number on their respective T-shirts, then in how many different ways these coins can be distributed ?

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

$\sum \limits_{ k =0}^6{ }^{51- k } C _3$ is equal to

If $^n{C_r} = 84,{\;^n}{C_{r - 1}} = 36$ and $^n{C_{r + 1}} = 126$, then $n$ equals