If $(1+x)^n=C_0+C_1 x+C_2 x^2+\ldots+C_n x^n$,then $C_0+2 C_1+3 C_2+\ldots+(n+1) C_n$ is equal to

If $A = \left\{ \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix} : a_i, b_i, c_i \in \{ \text{binomial coefficients in the expansion of } (1+x)^{11} \} \right\}$,then the number of elements in set $A$ is: (in $^9$)

$\sum_{k=0}^{20} \left({}^{20}C_{k}\right)^{2}$ is equal to :

The sum of the series $1 + \frac{1}{2} {}^{n}C_{1} + \frac{1}{3} {}^{n}C_{2} + \dots + \frac{1}{n+1} {}^{n}C_{n}$ is equal to

$\binom{n}{n-r} + \binom{n}{r+1}$,whenever $0 \le r \le n-1$,is equal to

The value of the sum $\left({ }^{n} C_{1}\right)^{2}+\left({ }^{n} C_{2}\right)^{2}+\left({ }^{n} C_{3}\right)^{2}+\ldots+\left({ }^{n} C_{n}\right)^{2}$ is