10626 different combinations.
This is a "Stars and Bars" type of problem with their being 20 "stars" and 5 bins which results in (5-1 = 4) "bars". The problem can be rephrased as "How many distinct values of x1,x2,x3,x4,x5 can you make for the following equation?"
x1 + x2 + x3 + x4 + x5 = 20
The general equation to this problem is
C=(n+k-1)!/(n!(k-1)!)
where
C = number of combinations
n = number of stars
k = number of bins
So substitute the known value and solve.
C=(20+5-1)!/(20!(5-1)!)
C=24!/(20!4!)
C= 24*23*22*21*20!/(20!4!)
C= 24*23*22*21/(4*3*2*1)
C= 24*23*22*21/24
C= 23*22*21/1
C= 23*22*21
C= 10626
