You need non-negative integer solutions to p+5n+10d+25q = 82.
If p = leftovers, then you simply need 5n + 10d + 25q ≤ 80.
So this is the same as n + 2d + 5q ≤ 16
So now you simply have to "crank out" the cases.
Case q=0 [ n + 2d ≤ 16 ]
Case (q=0,d=0) → n = 0 through 16 [17 possibilities] Case (q=0,d=1) → n = 0 through 14 [15 possibilities] ... Case (q=0,d=7) → n = 0 through 2 [3 possibilities] Case (q=0,d=8) → n = 0 [1 possibility]
Total from q=0 case: 1 + 3 + ... + 15 + 17 = 81
Case q=1 [ n + 2d ≤ 11 ] Case (q=1,d=0) → n = 0 through 11 [12] Case (q=1,d=1) → n = 0 through 9 [10] ... Case (q=1,d=5) → n = 0 through 1 [2]
Total from q=1 case: 2 + 4 + ... + 10 + 12 = 42
Case q=2 [ n + 2 ≤ 6 ] Case (q=2,d=0) → n = 0 through 6 [7] Case (q=2,d=1) → n = 0 through 4 [5] Case (q=2,d=2) → n = 0 through 2 [3] Case (q=2,d=3) → n = 0 [1]
Total from case q=2: 1 + 3 + 5 + 7 = 16
Case q=3 [ n + 2d ≤ 1 ] Here d must be 0, so there is only the case: Case (q=3,d=0) → n = 0 through 1 [2]
