# Integer solutions with subtraction

I just want to confirm if i have done this correctly.

$x_1 - x_2 + x_3 + x_4 + x_5 + x_6 = 107$

$x_1 \geq 0$
$x_2 \geq 0, x_{2} \leq 2$
$x_3 \geq 4$
$x_4 \geq 2$
$x_5 \geq 7$
$x_6 \geq 4$

Case 1: $x_2 = 0$
$x_1 + x_3 + x_4 + x_5 + x_6 = 90$ ie: $107 - 17$
$C (94, 4)$ possibilities

Case 2: $x_2 = 1$
$x_1 + x_3 + x_4 + x_5 + x_6 = 90$ ie (107 - 17 +1 from x2 = 1)
$C (95, 4)$ possibilities

Case 3: $x_2 = 2$
$C(96, 4)$ possibilities

Total num posibilities = $C(94,4) + C(95,4) + C(96,4)$

is this correct?

• It is indeed correct. – N. F. Taussig Dec 17 '17 at 23:04