How many ways to do choose $\leq 10$ from $5$ sets of $30$ objects. I have $5$ sets of letters each of size $30$ each. More specifically I have thirty 
'a's,'b's,'c's,'d's and 'e's.
How many ways can I choose to paint $10$ or less of them?
So I seem to have $x_1 + x_2 + x_3 + x_4 + x_5 \leq 10$, $x_i \in \mathbb{N}$
Set $x_1 \leq x_2 \leq x_3 \leq x_4 \leq x_5$
Is this the correct starting direction?
 A: Outline: We assume that the $30$ objects in each group are identical. 
The following is a useful trick. The number of non-negative integer solutions of $x_1+x_2+x_3+x_4+x_5\le 10$ is the number of non-negative integer solutions of $x_1+x_2+x_3+x_4+x_5+x_6=10$. 
The second problem  is a standard one, solved using "Stars and Bars."
Since $10$ is less than $31$, the constraint $x_i\le 30$ makes no difference.
Edit: We give a quick informal proof of the validity of the trick. I have $10$ candies, and want to give out $10$ or fewer of them to $5$ kids. The number of ways to do this is the number of solutions of $x_1+\cdots+x_5\le 10$.
Suppose we give any candies leftover from the $10$ to me. So effectively, we are distributing $10$ candies between $6$ "kids," one of them somewhat overaged. The number of ways to do this is the number of solutions of $x_1+\cdots+x_5+x_6=10$.
Somewhat more formally, there is a bijection between the distributions of $10$ or fewer to $5$ kids and the distributions of exactly $10$ to $6$ kids. 
