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

In the expansion of $(1+a)^{m+n},$ prove that the coefficients of $a^{m}$ and $a^{n}$ are equal.

Let $Q(x)$ be a polynomial of degree $n$. If $Q(1)=1$ and $\frac{Q(2x)}{Q(x+1)}+\frac{56}{x+7}-8=0$,then the value of ${}^nC_0+{}^nC_1+\ldots+{}^nC_n$ is equal to

Let $(1+x)^{10} = \sum_{r=0}^{10} c_{r} x^{r}$ and $(1+x)^{7} = \sum_{r=0}^{7} d_{r} x^{r}$. If $P = \sum_{r=0}^{5} c_{2r}$ and $Q = \sum_{r=0}^{3} d_{2r+1}$,then $\frac{P}{Q}$ is equal to:

If $3 \times { }^5 C_0 + 8 \times { }^5 C_1 + 13 \times { }^5 C_2 + 18 \times { }^5 C_3 + 23 \times { }^5 C_4 + 28 \times { }^5 C_5 = k \times 2^n$,where $n$ is a power of $2$,find $k$. Specifically,if $3 \times { }^5 C_0 + 8 \times { }^5 C_1 + 13 \times { }^5 C_2 + 18 \times { }^5 C_3 + 23 \times { }^5 C_4 + 28 \times { }^5 C_5 = k \times 2^4$,then $k=$

If $(1+x)^n = C_0 + C_1 x + C_2 x^2 + \ldots + C_n x^n$ for $n \in N$,then $C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \ldots + \frac{C_n}{n+1} =$