The ratio of the coefficients of the terms $x^{n-r}a^r$ and $x^ra^{n-r}$ in the binomial expansion of $(x+a)^n$ is:

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 $\frac{P}{2Q}$ is equal to

$\binom{50}{4} + \sum_{i=1}^{6} \binom{56-i}{3} = \dots$

If $\sum_{r=1}^{30} \frac{r^2({}^{30}C_r)^2}{{}^{30}C_{r-1}} = \alpha \times 2^{29}$,then $\alpha$ is equal to

The sum to $(n + 1)$ terms of the following series $\frac{C_0}{2} - \frac{C_1}{3} + \frac{C_2}{4} - \frac{C_3}{5} + \dots$ is

The value of $\sum_{r=0}^{6} \left({}^{6}C_{r} \cdot {}^{6}C_{6-r}\right)$ is equal to: