Permutations And Combinations — Practice Questions

For $1 \leq r \leq n$, the value of $^{n} C_{r}+^{n-1} C_{r}+\ldots .+^{r} C_{r}$ is

The value of $\left(^{7} C_{0}+^{7} C_{1}\right)+\left(^{7} C_{1}+^{7} C_{2}\right)+\ldots+\left(^{7} C_{6}+^{7} C_{7}\right)$ is

If $n$ is an even integer, then the value of $^{n} C_{0}+^{n} C_{2}+^{n} C_{4}+\ldots+^{n} C_{n}$ equals to

The total number of positive integral solutions $(x, y, z)$ such that $x y z=24$ is