If $\sum\limits_{i = 1}^{20} {\left( {\frac{{{}^{20}{C_{i - 1}}}}{{{}^{20}{C_i} + {}^{20}{C_{i - 1}}}}} \right)} ^3 = \frac{k}{21}$,then $k$ equals

If $n$ is a positive integer greater than $1$,then $3({ }^n C_0) - 8({ }^n C_1) + 13({ }^n C_2) - 18({ }^n C_3) + \ldots$ up to $(n+1)$ terms $=$

Find the arithmetic mean of $^nC_0, ^nC_1, ^nC_2, \dots, ^nC_n$.

The sum of the last eight coefficients in the expansion of $(1 + x)^{15}$ is

If $\sum\limits_{K = 1}^{12} {12K \cdot {^{12}C_K} \cdot {^{11}C_{K - 1}}} $ is equal to $\frac{{12 \times 21 \times 19 \times 17 \times \dots \times 3}}{{11!}} \times {2^{12}} \times p$,then $p$ is:

If for $3 \leq r \leq 30$, $\binom{30}{30-r} + 3\binom{30}{31-r} + 3\binom{30}{32-r} + \binom{30}{33-r} = \binom{m}{r}$, then $m$ equals: