I am asked to find how many integral solutions there..

x1+x2+x3+..........+x10=27

It asked ...

a.if all x>=1

b.if x1 ≥ 2, x2 ≥ 3, and the remaining xi ≥ 0

c.if all xi ≥ 1, and x1 ≤ 16

d.if all xi ≥ 1, and x1 ≤ 3 and x2 ≤ 2

now for the first one it was easy I think

26 choose 9

But once it is >=0 I get a little confused.

First say it said "all Xi>=0"

Is the formula "n+k-1 choose k"?

Then for question b. would I go...

Y1=X1-1

Y2=X2-2

Then +1 for each of the remaining xi which is +8?

So for b it would be

27-1-2+8=32

So...

b. 32 choose 9?

For

c.

Would it be 26 choose 9 subtract 10 choose 9 (number of solution with x1>16)