Multiple Sums

There are 12 different ways you can add two numbers, between 1 and 25,
to total 25:
(1+24, 2+23, ..., 12+13).
No number is used more than once. How many different combinations using
3 numbers to total 25 are there? Remember, use the numbers only once for
each grouping. (i.e. 25 = 1+2+22 = 1+4+20 = 2+4+19, but not 1+12+12)

Can you fill a table for the number of possible sums for totals
from 20 to 30, with the number of terms
in each sum from 2 to 7? Can you find a formula which would provide the
answer for the number of possible sums of N, with k terms in each sum?
Source: Mind Bending Puzzles Calendar,
Terry Stickles, January 14, 2000.
Solution
Mail to Ken