Constrained integer partition Challenge
In this challenge, all numbers are in \$\mathbb{N}_0\$.
Create a function or program that, when given a number \$N\$ and a tuple of \$k\$ numbers \$(n_i)\$ (all ≤ \$N\$), returns the number of ways \$N\$ can be written as a sum of \$k\$ integers (\$x_1 + x_2 + ... + x_k\$) such that \$n_i \le x_i \le N\$.
The input format is not fixed. You can read and parse strings, take two parameters as int and int[], etc.
This is a variation of the classical Integer Partition problem.
Test Cases
\$N=4, n=(0, 0, 2) \implies 6\$ (2+0+2, 1+1+2, 0+2+2, 1+0+3, 0+1+3, 0+0+4)
\$N=121, n=(7, 16, 21, 36, 21, 20) \implies 1\$ (7+16+21+36+21+20)
This is code-golf, so the lowest byte count for each language wins!
Jelly, 5 bytes
rŒp§ċ

A dyadic Link accepting the lower bound tuple on the left and the total on the right which yields the count.
Try it online!
How?
rŒp§ċ - Link: T, N       e.g. [2,1,3], 5
r     - inclusive range       [[2,3,4,5],[1,2,3,4,5],[1,2,3,4,5]]
 Œp   - Cartesian product     [[2,1,1],[2,1,2],[2,1,3],[2,1,4],[2,1,5],[2,2,1],...,[5,5,4],[5,5,5]]
   §  - sums                  [ 4,      5,      6,      7,      8,      5,     ..., 14,     15]
    ċ - count (Ns)            2