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

A set contains $(2n + 1)$ elements. The number of sub-sets of the set which contains at  most $n$ elements is :-

Two packs of $52$ cards are shuffled together. The number of ways in which a man can be dealt $26$ cards so that he does not get two cards of the same suit and same denomination is

The number of onto functions $f$ from $\{1, 2, 3, …, 20\}$ only $\{1, 2, 3, …, 20\}$ such that $f(k)$ is a multiple of $3$, whenever $k$ is a multiple of $4$, is

$^n{C_r}{ + ^{n - 1}}{C_r} + ......{ + ^r}{C_r}$ =

If for some $\mathrm{m}, \mathrm{n} ;{ }^6 \mathrm{C}_{\mathrm{m}}+2\left({ }^6 \mathrm{C}_{\mathrm{m}+1}\right)+{ }^6 \mathrm{C}_{\mathrm{m}+2}>{ }^8 \mathrm{C}_3$ and ${ }^{n-1} P_3:{ }^n P_4=1: 8$, then ${ }^n P_{m+1}+{ }^{n+1} C_m$ is equal to