Constrained integer partition
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