If $x,\;y$ and $r$ are positive integers, then $^x{C_r}{ + ^x}{C_{r - 1}}^y{C_1}{ + ^x}{C_{r - 2}}^y{C_2} + .......{ + ^y}{C_r} = $

A committee of $7$ has to be formed from $9$ boys and $4$ girls. In how many ways can this be done when the committee consists of:

at least $3$ girls?

The least value of a natural number $n$ such that $\left(\frac{n-1}{5}\right)+\left(\frac{n-1}{6}\right) < \left(\frac{n}{7}\right)$, where $\left(\frac{n}{r}\right)=\frac{n !}{(n-r) ! r !}, i$

The number of ways of dividing $52$ cards amongst four players equally, are

Statement$-1:$ The number of ways of distributing $10$ identical balls in $4$ distinct boxes such that no box is empty is $^9C_3 .$

Statement$-2:$ The number of ways of choosing any $3$ places from $9$ different  places is $^9C_3 $.

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