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

$n$ balls each of weight $w$ when weighted in pairs the sum of the weights of all the possible pairs is $120$ when they are weighed in triplets the sum of the weights comes out to be $480$ for all possible triplets, then $n$ is

At an election, a voter may vote for any number of candidates, not greater than the number to be elected. There are $10$ candidates and $4$ are of be selected, if a voter votes for at least one candidate, then the number of ways in which he can vote is

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

If $^8{C_r}{ = ^8}{C_{r + 2}}$, then the value of $^r{C_2}$ is