# How many integer solutions are there to \$x_1+x_2+x_3+x_4+x_5=31\$

with
a) $$x_igeq0$$

b) $$x_i>0$$

c) $$x_i geq i(i = 1, 2, 3, 4, 5)$$

Now $$a)$$ seems simple enough to me, just use stars and bars to have $$C(31+5-1, 31) = 52360$$

But I’m completely stuck on b) and c), I’m not sure what do to with these constraints.

Any tips would be extremely appreciated.