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Find a formula for the number of solutions to $$ x_1 + x_2 + x_3 + \ldots + x_k = n $$ where $n \ge 0$ and the the $x_i$ are non-negative integers. For instance, if $n > 0$ then there is exactly one solution to $x_1 = n$. There are $n + 1$ solutions to $x_1 + x_2 = n$. How many solutions to the equation $x_1 + x_2 + x_3 = n$ are there when $k = 3$? Make a table of values for various $n$ and $k$ and generalize your answer for $k > 3$.

I'm not sure how to start on this question but it's somehow related to n!

FYY
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anonymous9254
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1 Answers1

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Hint: there is an interesting interpretation of your question: Suppose, for instance, $k=3$ and $n=5$. Then draw $n$ balls and $k-1$ walls, in some order: $$ \circ\circ\mid\circ\mid\circ\circ $$ This can be interpreted as the number 5 written as $2+1+2$. Using this idea, can you solve the question?

FYY
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