If ${ }^{12} C_{2 k-1}={ }^{12} C_{k+1}$,then find $k$.

$A$ student has to answer $10$ questions,choosing at least $4$ from each of the parts $A$ and $B$. If there are $6$ questions in part $A$ and $7$ in part $B$,then the number of ways the student can choose $10$ questions is:

The domain of the function $f(x) = {}^{(10-x)}C_{(x-6)}$ is...

$A$ father with $8$ children takes $3$ children to the zoo at a time. If he can go more than once without taking the same $3$ children together,in how many ways can the father go to the zoo?

$A$ committee of $12$ members is to be formed from $9$ women and $8$ men. The number of committees in which the women are in majority is

If ${}^{2n}C_3 : {}^{n}C_3 = 10 : 1$,then the ratio $(n^2 + 3n) : (n^2 - 3n + 4)$ is