Stars and Bars (Hard)

Stars and Bars: core idea

• 1. Partitions of $12$ into exactly $4$ parts. How many?

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• 2. Surjective functions $[8]\to[3]$. How many?

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• 3. Positive integers $x\le y\le z$ with $\frac1x+\frac1y+\frac1z=1$. Count triples.

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• 4. Catalan $C_7$.

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• 5. Nonnegative solutions to $x_1+2x_2+3x_3=30$.

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• 6. Choose $5$ from $\{1,\ldots,20\}$ with no two consecutive. How many?

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• 7. Coefficient of $x^{15}$ in $\prod_{k=1}^{5}(1+x^k+x^{2k}+\cdots)$.

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• 8. Positive $6$-tuples summing to $20$ with $\gcd=1$. Count?

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• 9. Ways to write $20$ as ordered sum of powers of $2$ (parts $\ge1$)?

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