How to compute combinations with repetitions (stars and bars)?

**Stars and Bars Explanation:** We want to choose $k$ identical items from $n$ types, allowing repetition. Represent the $k$ items as stars ($*$), and use $n-1$ bars ($|$) to separate the $n$ types. For example, choosing 10 donuts from 5 flavors: arrange 10 stars and 4 bars in a row. The number of stars before the first bar = number of flavor 1 donuts, between bar 1 and 2 = flavor 2, etc. Total arrangements = $\binom{10 + 5 - 1}{10} = \binom{14}{10} = 1001$. **General Formula:** $$\binom{k + n - 1}{k} = \binom{k + n - 1}{n - 1}$$ **Applying to Equations:** For $x_1 + x_2 + x_3 + x_4 = 15$ with $x_i \geq 0$: - $n = 4$ types, $k = 15$ items - Solutions: $\binom{15 + 4 - 1}{15} = \binom{18}{3} = 816$ For **positive** solutions $x_i \geq 1$: Let $y_i = x_i - 1 \geq 0$. Then $y_1 + y_2 + y_3 + y_4 = 15 - 4 = 11$. Solutions: $\binom{11 + 4 - 1}{11} = \binom{14}{3} = 364$ **Key Insight:** The stars and bars method converts combinatorial selection with repetition into counting arrangements of identical objects and dividers. This is one of the most powerful tools in combinatorics.