I am interested in how many non-negative integer solutions there are to:
$$x_1 + \ldots + x_N = B$$
where at least $K$ of the variables $x_1, \ldots , x_N \geq C$
For example when: $B = 5, N = 3, K = 2, C = 2$
I want to count the solutions to:
$$x_1 + x_2 + x_3 = 5$$
where at least $2$ of the variables are $\geq 2$.
I found the total number of candidate solutions using the $\binom{B+N-1}{B} = 21$
However, only $9$ of them have two variables $\geq 2$.
\begin{align*} 2+0+3& =5\\ 2+1+2& =5\\ 3+0+2& =5\\ 1+2+2& =5\\ 3+2+0& =5\\ 0+2+3& =5\\ 0+3+2& =5\\ 2+3+0& =5\\ 2+2+1& =5 \end{align*}
I feel there is a connection to the Associated Stirling numbers of the second kind. But I can't place it :(
EDIT:
Here is my code for enumerating them all to count the number of ways of select B elements from a set of N (uniformly with replacement), such that you have at least C copies of K elements - also shows the output for this question I'm asking here as it's the core piece. Obviously can't be run for very large values of the parameters - that's why I'm here :) Code is here
Here is another example for B = 6, N = 3, C = 2 and K = 2 there are 16 solutions:
\begin{align*} 0+2+4& = 6\\ 0+3+3& = 6\\ 0+4+2& = 6\\ 1+2+3& = 6\\ 1+3+2& = 6\\ 2+0+4& = 6\\ 2+1+3& = 6\\ 2+2+2& = 6\\ 2+3+1& = 6\\ 2+4+0& = 6\\ 3+0+3& = 6\\ 3+1+2& = 6\\ 3+2+1& = 6\\ 3+3+0& = 6\\ 4+0+2& = 6\\ 4+2+0& = 6\\ \end{align*}
There are a number of different and correct solutions below. I don't know which to accept.