If $n$ is a positive integer,then $\sum_{r=1}^n r \cdot C_r =$

If $a$ and $d$ are two complex numbers,then the sum to $(n + 1)$ terms of the following series $a{C_0} - (a + d){C_1} + (a + 2d){C_2} - \dots$ is

$C_0 - C_1 + C_2 - C_3 + \dots + (-1)^n C_n$ is equal to

$\sum\limits_{r = 0}^m {^{n + r}{C_n} = } $

For $2 \le r \le n$,$\binom{n}{r} + 2\binom{n}{r-1} + \binom{n}{r-2}$ is equal to

If $\sum\limits_{i = 0}^4 {^{4 + i}} {C_i} + \sum\limits_{j = 6}^9 {^{3 + j}} {C_j} = {\,^x}{C_y}$ ($x$ is a prime number),then which one of the following is incorrect?