The number of distinct prime divisors of the number $(512)^3 - (253)^3 - (259)^3$ is

Let $(1 + x)(1 + x + x^2)(1 + x + x^2 + x^3) \dots (1 + x + x^2 + \dots + x^{30}) = a_0 + a_1x + a_2x^2 + \dots + a_{465}x^{465}$. Then the sum $a_0 + a_2 + a_4 + \dots$ 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?

The coefficient of $x^{2012}$ in the expansion of $(1-x)^{2008}(1+x+x^2)^{2007}$ is equal to

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:

Let the smallest value of $k \in N$, for which the coefficient of $x^3$ in $(1+x)^3 + (1+x)^4 + \dots + (1+x)^{99} + (1+kx)^{100}, x \neq 0$, is $(43n + \frac{101}{4}) ({}^{100}C_3)$ for some $n \in N$, be $p$. Then the value of $p+n$ is: