Confusion regarding the method to solve combinations with repetition Looking at some examples and the methods involved in solving combinations with repetition, for example with the problem, how many distinct solutions are there to:
a + b = 10, where a>=0

Intuitively, well there are 6 solutions:
0 + 10 
1 + 9
2 + 8
3 + 7
4 + 6
5 + 5

I have left out 6 + 4 etc.. As order does not matter so these arrangements will not count.
Now using the formula of combinations with repetition:
(n + r − 1)! / r!(n − 1)!  =>  11! / 10! = 11

I get 11 combinations, so the formula is including what I had left out ie (6 + 4) .... (10 + 0). But wouldn't this now mean that order does matter?
 A: You have to remember why the $a+b=10$ problem is relevant to counting "combinations with repetition." Let's look at a bigger example, say, counting integer solutions $a+b+c+d=8,$ with $a,b,c,d\ge 0.$ The reason we care about solutions to this equation is that they represent ways to select $8$ items from a pool of $4$ items, where repetition is allowed and order of the selected item does not matter. The variable $a$ represents the number of times the first item is selected, $b$ corresponds to the second item, etc. Therefore, whether order matters or not depends on which lens you are looking at the problem with.

*

*If you are thinking of the solution in terms of quadruples $(a,b,c,d)$ such that $a+b+c+d=8$, then order does matter. $(3,2,2,1)$ is different from $(2,3,2,1)$.


*If you are thinking of the solution in terms of combinations with repetition, then order does not matter. If the set being chosen from is $\{A,B,C,D\}$, the $(3,2,2,1)$ corresponds to $\{A,A,A,B,B,C,C,D\}$, which would be the same as $\{A,B,C,A,B,C,D,A\}$.
A: Notice that $b$ is fully determined by $a,$ so each distinct solution $(a_k,b_k)$ is just $(a_k,10-a_k).$
In other words, notice that the question can be equivalently rephrased as "How many distinct non-negative integer values of $a$ satisfy $a+b=10$?"
Thus, there are indeed 11 distinct solutions, where $k\in\{0,2,\ldots,10\}$ and $a_k=k.$
