MACM 201: counting integer solutions

Original equation:
x1 + x2 + x3 + x4 = 21

Original conditions:
-1 <= x1 <= 3
2 <= x2 <= 5
-3 <= x3 <= 7
5 <= x4 <= 11

Substitutions:
x1 = y1 - 1
x2 = y2 + 2
x3 = y3 - 3
x4 = y4 + 5

Modified equation:
(y1 - 1) + (y2 + 2) + (y3 - 3) + (y4 + 5) = 21

Simplified equation:
y1 + y2 + y3 + y4 = 18

Modified conditions:
-1 <= (y1 - 1) <= 3
2 <= (y2 + 2) <= 5
-3 <= (y3 - 3) <= 7
5 <= (y4 + 5) <= 11

Simplified conditions:
0 <= y1 <= 4
0 <= y2 <= 3
0 <= y3 <= 10
0 <= y4 <= 6

Complemented conditions:
c1: y1 >= 5
c2: y2 >= 4
c3: y3 >= 11
c4: y4 >= 7

k - 1 = 3

N = C(21, 3) = 1330
N(c1) = C(16, 3) = 560
N(c2) = C(17, 3) = 680
N(c3) = C(10, 3) = 120
N(c4) = C(14, 3) = 364
N(c1c2) = C(12, 3) = 220
N(c1c3) = C(5, 3) = 10
N(c1c4) = C(9, 3) = 84
N(c2c3) = C(6, 3) = 20
N(c2c4) = C(10, 3) = 120
N(c3c4) = C(3, 3) = 1
N(c1c2c3) = C(1, 3) = 0
N(c1c2c4) = C(5, 3) = 10
N(c1c3c4) = C(-2, 3) = 0
N(c2c3c4) = C(-1, 3) = 0
N(c1c2c3c4) = C(-6, 3) = 0

  ________
N(c1c2c3c4) = N - N(c1) - N(c2) - N(c3) - N(c4) + N(c1c2) + N(c1c3) + N(c1c4) +
N(c2c3) + N(c2c4) + N(c3c4) - N(c1c2c3) - N(c1c2c4) - N(c1c3c4) - N(c2c3c4) + N(
c1c2c3c4)

= 51

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