In the expansion of $(x-2y+3z)^5$,if the total number of terms is $p$ and the coefficient of $x^2yz^2$ is $q$,then $\frac{q}{p}=$

The coefficient of $x^{1012}$ in the expansion of $(1 + x^n + x^{253})^{10}$,where $n \leq 22$ is any positive integer,is

Let $N$ be the least positive integer such that whenever a non-zero digit $c \in \{1, 2, \dots, 9\}$ is written after the last digit of $N$,the resulting number is divisible by $c$. The sum of the digits of $N$ is:

The sum of all positive divisors of $960$ is

$A$ natural number $n$ such that $n!$ ends in exactly $1000$ zeros is

Three players play a total of $9$ games. In each game,one person wins and the other two lose; the winner gets $2$ points and the losers get $-1$ each. The number of ways in which they can play all the $9$ games and finish each with a zero score is