The coefficient of $x^{13}$ in the expansion of $(1 - x)^5(1 + x + x^2 + x^3)^4$ is :-

For the natural numbers $m, n$,if $(1-y)^{m}(1+y)^{n}=1+a_{1} y+a_{2} y^{2}+\ldots +a_{m+n} y^{m+n}$ and $a_{1}=a_{2}=10$,then the value of $(m+n)$ is equal to:

If $a_r$ is the coefficient of $x^r$ in the expansion of $(1 + x + x^2)^n$,then $a_1 - 2a_2 + 3a_3 - \dots - 2n\,a_{2n} = $

Let $a_1, a_2, a_3, \dots, a_{100}$ be positive real numbers and $S_k$ be the sum of products of $a_1, a_2, \dots, a_{100}$ taken $k$ at a time. If $S_{98} S_2 \ge \lambda (a_1 a_2 \dots a_{100})$,then $\lambda$ is

$\sum_{\substack{i, j=0 \\ i \neq j}}^{n} {}^{n}C_{i} {}^{n}C_{j}$ is equal to

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?