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

Let the coefficient of $x^{r}$ in the expansion of $(x+3)^{n-1}+(x+3)^{n-2}(x+2)+(x+3)^{n-3}(x+2)^2+\ldots+(x+2)^{n-1}$ be $\alpha_{r}$. If $\sum_{r=0}^{n-1} \alpha_{r}=\beta^{n}-\gamma^{n}$,where $\beta, \gamma \in N$,then the value of $\beta^2+\gamma^2$ equals:

If $(1 + x - 3x^2)^{2145} = a_0 + a_1x + a_2x^2 + \dots$,then $a_0 - a_1 + a_2 - a_3 + \dots$ ends with:

Ten trucks,numbered $1$ to $10$,are carrying packets of sugar. Each packet weighs either $999 \ g$ or $1000 \ g$,and each truck carries only packets of equal weight. The combined weight of $1$ packet selected from the first truck,$2$ packets from the second,$4$ packets from the third,and so on,up to $2^9$ packets from the tenth truck is $1022870 \ g$. Which trucks are carrying the lighter bags?

The fractional part of a real number $x$ is defined as $x - [x]$,where $[x]$ is the greatest integer less than or equal to $x$. Let $F_1$ and $F_2$ be the fractional parts of $(44 - \sqrt{2017})^{2017}$ and $(44 + \sqrt{2017})^{2017}$,respectively. Then,$F_1 + F_2$ lies between the numbers:

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 $m, n$