mathematical equation needed?

Question

If the sum of a number of parameters (x) equals (y).
What is the equation that provides me with the number of combinations of values for each parameter of x?

e.g. x = 2 and y = 10
So we have 2 parameters that when added together give 10.
How many sets of numbers are valid for this?
The answer here is 11 - but what is a generic equation?
(0,10; 1,9; 2,8; 3,7; 4,6; 5,5; 6,4; 7,3; 8,2; 9,1; 10,0)

Answer 1

There is a formula for this.

I can explain it but it would take a while. If you want to find an explanation on the net, it's the same formula as you would use to distribute y indistinguishable objects into x boxes.

The formula is: C(x + y - 1, y), or (x + y - 1)!/(y!(x + y - 1 - y)!

In your example:

C(2 + 10 - 1, 10) = C(11, 10) = C(11, 1) = 11

As another example, let x be 3 and y by 5.

The combinations would be:

(0, 0, 5), (0, 1, 4), (0, 2, 3), (0, 3, 2), (0, 4, 1), (0, 5, 0),
(1, 0, 4), (1, 1, 3), (1, 2, 2), (1, 3, 1), (1, 4, 0),
(2, 0, 3), (2, 1, 2), (2, 2, 1), (2, 3, 0),
(3, 0, 2), (3, 1, 1), (3, 2, 0),
(4, 0, 1), (4, 1, 0),
(5, 0, 0)

That's 21 different ways.

From the formula: C(3 + 5 - 1, 5) = C(7, 5) = C(7,2) = 7x6/2 = 21

Here's an explanation of the formula....not the best ...looking for a better one:

http://www.math.northwestern.edu/~mlerma/courses/cs310-04w/notes/dm-morecomb.pdf

Here's a better one:

http://www.csee.umbc.edu/~stephens/203/PDF/6-5.pdf

Answer 2

x + y =10