$^nC_r + ^{n-1}C_r + ...... + ^rC_r =$

The product of any $r$ consecutive natural numbers is always divisible by

The value of ${ }^{49} C_3+{ }^{48} C_3+{ }^{47} C_3+{ }^{46} C_3+{ }^{45} C_3+{ }^{45} C_4$ is

$A$ student is asked to answer $10$ out of $13$ questions in an examination such that he must answer at least four questions from the first five questions. The number of choices available to him is

$A$ father with $8$ children takes them $3$ at a time to the Zoological gardens,as often as he can without taking the same $3$ children together more than once. The number of times each child will go to the garden is

If $\binom{n}{3} + \binom{n}{4} > \binom{n+1}{3}$,then: