Can you help me to find the solution to the following question?
Fifty identical dice are rolled. How many distinct outcomes with positive even number of 1’s, 2’s, ..., of 6’s are possible? For example, 8 of them 1, 14 of them are 2, 10 of them are 3, 10 of them are 4, 2 of them are 5, 6 of them are 6.
I am trying to write 50 as sum of even numbers such that $50 = 50 + 0$, $50 = 48 + 2$, $50 = 46 + 4$, $50 = 46 + 2 + 2$, etc.
By using this method, I can solve this question. I am curious to know about any other methods.
I assume that you do not care about the order in which the numbers are rolled... Any way of rolling twenty $2$'s and thirty $3$'s will be considered the same, whether the $2$'s appeared first or the $3$'s appeared first or they appeared mixed.
Letting $x_i$ represent the number of times that the number $i$ was rolled for each $i\in\{1,2,3,4,5,6\}$ respectively, you are trying to count the number of non-negative integer solutions to $\begin{cases}x_1+x_2+x_3+x_4+x_5+x_6 = 50\\0\leq x_i\\x_i~\text{even}\end{cases}$
As for a hint... recall that if $x_i$ is even then it can be written as $x_i=2y_i$ for some $y_i$.
Further, Stars-and-bars is a common technique/explanation for how to proceed.