How many solutions (using only nonnegative integers) are there to the following equation? x1 + x2 + x3 + x4 + x5 = 30

Question

How many solutions (using only nonnegative integers) are there to the following equation?   x1 + x2 + x3 + x4 + x5 = 30

anonymous · Answer

So trying to figure out the math here I know manually I can do 30= 30+0+0+0+0 and so on but that will take for ever

anonymous · Answer

Let's call this notched line a fence. Say we want to figure out how many ways we can color 5 different segments. We want to color one picket at a time (there are 30 total), and we want pickets that are colored the same to be grouped together.

Finding out the total number of color schemes is the same as this problem, where each integer $x_i$ counts the number of pickets of the $i$-th color.

Here's an example:
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