If $1 + (2 + {}^{49}C_{1} + {}^{49}C_{2} + \dots + {}^{49}C_{49})({}^{50}C_{2} + {}^{50}C_{4} + \dots + {}^{50}C_{50})$ is equal to $2^{n} \cdot m$,where $m$ is odd,then $n + m$ is equal to.

Let $\alpha = \sum_{r=0}^{n} (4r^2+2r+1) {}^{n}C_{r}$ and $\beta = \left(\sum_{r=0}^{n} \frac{{}^{n}C_{r}}{r+1}\right) + \frac{1}{n+1}$. If $140 < \frac{2\alpha}{\beta} < 281$,then the value of $n$ is...............

For non-negative integers $s$ and $r$,let $\binom{s}{r} = \begin{cases} \frac{s!}{r!(s-r)!} & \text{if } r \leq s \\ 0 & \text{if } r > s \end{cases}$. For positive integers $m$ and $n$,let $g(m, n) = \sum_{p=0}^{m+n} \frac{f(m, n, p)}{\binom{n+p}{p}}$,where for any non-negative integer $p$,$f(m, n, p) = \sum_{i=0}^{p} \binom{m}{i} \binom{n+i}{p} \binom{p+n}{p-i}$. Then which of the following statements is/are $TRUE$?
$(A)$ $g(m, n) = g(n, m)$ for all positive integers $m, n$
$(B)$ $g(m, n+1) = g(m+1, n)$ for all positive integers $m, n$
$(C)$ $g(2m, 2n) = 2g(m, n)$ for all positive integers $m, n$
$(D)$ $g(2m, 2n) = (g(m, n))^2$ for all positive integers $m, n$

The coefficient of $t^{20}$ in the expansion of $(1 + t^2)^{10}(1 + t^{10})(1 + t^{20})$ is

The coefficient of $t^{24}$ in the expansion of $(1 + t^2)^{12}(1 + t^{12})(1 + t^{24})$ is

If $(1+x+x^2+x^3)^5 = \sum_{k=0}^{15} a_k x^k$, then $\sum_{k=0}^7 (-1)^k a_{2k}$ is equal to