Category Archives: Combinatorics

Sum and Product of All Factors

Posted on December 18, 2020 by kevin

For a nature number $$n=p_{1}^{q_{1}}\times p_{2}^{q_{2}}\times \cdots \times p_{m}^{q_{m}}$$ where $p_{i} (1 \le i \le m) $ are unique prime numbers. The total number of positive factors of $n$ is $$f=(q_{1}+1)\times(q_{2}+1)\times\cdots\times(q_{m}+1)$$ The sum of all positive factors of $n$ is … Continue reading →

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Cheat Sheet for Distributing $k$ Balls into $n$ Boxes

Posted on November 19, 2020 by kevin

Distribution of Restriction $k\text{ Balls}$ $n\text{ Boxes}$ None $\le 1$ $\ge 1$ $=1$ $distinct$ $distinct$ $$n^k$$ $$(n)_k$$ $$n!S(k,n)$$ $$n!\text{ or }0$$ $identical$ $distinct$ $${n+k-1}\choose k$$ $$n\choose k$$ $${k-1}\choose{n-1}$$ $$1\text{ or }0$$ $distinct$ $identical$ $$\sum_{i=1}^{n}S(k,i)$$ $$1\text{ or }0$$ $$S(k,n)$$ $$1\text{ or … Continue reading →

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Coloring a Cube

Posted on May 3, 2020 by kevin

We worked on two different cube coloring problems before. One is to paint a unit cube with $1$ face in red, $1$ face in green, $1$ face in yellow, and $3$ faces in blue color. The other is to paint … Continue reading →

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MATHCOUNTS Exercises – 19

Posted on January 31, 2020 by kevin

For the following equation $$ x\times y\times z = 360$$ Determine the number of unique positive integer solutions. Determine the number of unique integer solutions. Click here for the solutions. Solution for Question 1 Because the product of $x$, $y$ … Continue reading →

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MATHCOUNTS Exercises – 18

Posted on January 30, 2020 by kevin

How many 4-digit integers between 1000 and 9999 have their digits in non-decreasing order, such as 3447? Click here for solutions. Solution 1 – Brute Force Denote $f(m, n)$ as the number of $m$-digit integers with their digits in non-decreasing … Continue reading →

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