Words of length $10$ are formed using the letters, $A, B, C, D, E, F, G, H, I, J$. Let $x$ be the number of such words where no letter is repeated ; and let $y$ be the number of such words where exactly one letter is repeated twice and no other letter is repeated. Then, $\frac{y}{9 x}=$

In an election the number of candidates is $1$ greater than the persons to be elected. If a voter can vote in $254$ ways, then the number of candidates is

What is the number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these

cards are of the same colour?

For a scholarship, atmost $n$ candidates out of $2n+1$ can be selected. If the number of different ways of selection of atleast one candidate for scholarship is $63$, then maximum number of candidates that can be selected for the scholarship is -

$\left( {\begin{array}{*{20}{c}}n\\{n - r}\end{array}} \right)\, + \,\left( {\begin{array}{*{20}{c}}n\\{r + 1}\end{array}} \right)$, whenever $0 \le r \le n - 1$is equal to

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