How do I find the number of combinations of numbers that add to give a value?

You could use generating functions

(z + z² + z³ + ... + z^10)^4

= [z (1 - z^10)/(1 - z)]^4

= z^4 (1 - 4 z^10 + 6 z^20 - 4 z^30 + z^40) (1 + z + z² + ...)^4

We use (1 + z + z² + ...)^n = 1 + C(n,1) z + C(n+1,2) z² + C(n+2,3) z³ + ...

= z^4 (1 - 4 z^10 + 6 z^20 - 4 z^30 + z^40) (1 + C(4,1) z + C(5,2) z² + C(6,3) z³ + .... )

We need the coefficient of z^33

we have z^4 in front, so we still need z^29 :

C(32,29) - 4 C(22,19) + 6 C(12,9) = 4960 - 6160 + 1320 = 120

So there are 120 solutions to the first example equation you gave.

I missed the part where the number must be <= 10. My answer only covers the part without that constraint.

This is a standard "stars and bars" problem. It gets that name from the way it's solved.

How many solutions are there to a+b+c+d = 33, where a b c and d are all >= 1?

Line up 33 stars. Divide them into sets by putting 3 bars into the 32 spaces between them. You can do that in 32C3 ways, since there are that many ways to choose the 3.

32C3 = 32x31x30 / 6 = 4960

EDIT: Yes, I agree with the thumbs down. I misread the question, so this isn't helpful. Sorry.

I did write a program to check the answer, and 120 is indeed correct.

Edit 2.

This isn't a general solution, but for 33, there are only 11 possible combinations for a,b,c, and d. You can list them, then count the number of permutations for each and add them up. I did that, and here's how you get 120:

10 10 10 3 with 4 permutations

10 10 9 4 with 12 permutations

10 10 8 5 with 12 permutations

10 10 7 6 with 12 permutations

10 9 9 5 with 12 permutations

10 9 8 6 with 24 permutations

10 9 7 7 with 12 permutations

10 8 8 7 with 12 permutations

9 9 9 6 with 4 permutations

9 9 8 7 with 12 permutations

9 8 8 8 with 4 permutations

Sum of the permutations is 120.