I was solving a combinatorics problem when I encountered difficulties. The problem was:
$x_1 \in \{0,1\}$
$x_2 \in \{0,1,2\}$
. .
$x_{n-1}\in\{0,1,2..,n-1\}$
We have to find the number of ways we can choose $x_1,x_2..x_{n-1}$ such that
$x_1+x_2+..+x_{n-1}=k$
$Given \ k\in\mathbb N \ and \ k\leq \frac{n(n-1)}{2}$
I approached this problem by developing a generating function.
$(t^0+t^1)(t^0+t^1+t^2)...(t^0+t^1+...+t^{n-1})$
Which solves to
$\frac{(1-t^2)(1-t^3)...(1-t^n)}{(1-t)^{n-1}}$
I planned to find the coefficient of $t^k$ in the expression above but i just cannot find a way. I dont know where I am not able to catch the pattern or something. I looked it up, and I stumbled upon q factorials whose definition seemed to be precisely the generating function I have encountered above. My aim remains the same, to calculate the coefficient of $t^k$ in the above generating function, which you may consider to be the coefficient of $q^k$ in the q factorial of $n$, thus effectively solving the original problem. Please help me find the coefficient and if you have an alternative approach to the original problem, that would be welcome! Thanks!
For reference : http://mathworld.wolfram.com/q-Factorial.html