Suppose $\sum_{r=0}^{2023} r \cdot ^{2023}C_r = 2023 \times \alpha \times 2^{2022}$. Then the value of $\alpha$ is $............$

Let $(1+2x)^{20} = a_0 + a_1x + a_2x^2 + \dots + a_{20}x^{20}$. Then $3a_0 + 2a_1 + 3a_2 + 2a_3 + 3a_4 + 2a_5 + \dots + 2a_{19} + 3a_{20}$ equals

If $n$ is a positive integer,then $\sum_{r=1}^n r^2 \cdot C_r = (\ldots \ldots \ldots) 2^{n-2}$

$^{4n}C_0 + ^{4n}C_4 + ^{4n}C_8 + ... + ^{4n}C_{4n}$ is

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

Let $c_0, c_1, c_2, \ldots, c_n$ be the binomial coefficients in the expansion of $(1+x)^n$. If $S_{n+1} = 5 \cdot c_0 + 8 \cdot c_1 + 11 \cdot c_2 + \ldots$ ($n+1$ terms),then $S_{11} =$