148 lines
5.0 KiB
Markdown
148 lines
5.0 KiB
Markdown
---
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title: Combinatorics
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TARGET DECK: Obsidian::STEM
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FILE TAGS: combinatorics set
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tags:
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- combinatorics
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- set
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---
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## Overview
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When selecting objects, we can use the given table to hint at what counting strategy we should use:
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Order | Repeats | Answer Shape | Reference
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----- | ------- | ------------------ | ---------
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Yes | Yes | $n^k$ | `-`
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Yes | No | $(n)_k$ | [[permutations#Falling Factorials]]
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No | Yes | $\binom{n + k}{k}$ | [[combinations#Stars and Bars]]
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No | No | $\binom{n}{k}$ | [[combinations]]
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%%ANKI
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Basic
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What does it mean for order to matter?
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Back: We get different outcomes if the same objects are selected in different orders.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708715147778-->
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END%%
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%%ANKI
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Basic
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What does it mean for repeats to be allowed?
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Back: The same object can be selected multiple times.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1708715147781-->
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END%%
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%%ANKI
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Basic
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What combinatorial *notation* corresponds to the highlighted square?
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![[ordering-y-repetition-y.jpg]]
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Back: $n^k$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709305803508-->
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END%%
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%%ANKI
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Basic
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What combinatorial *concept* corresponds to the highlighted square?
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![[ordering-y-repetition-y.jpg]]
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Back: The multiplicative principle.
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709305803515-->
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END%%
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%%ANKI
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Basic
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Which square corresponds to notation $n^k$?
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![[ordering-repetition.jpg]]
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Back:
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![[ordering-y-repetition-y.jpg]]
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<!--ID: 1709305803518-->
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END%%
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%%ANKI
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Basic
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What combinatorial *notation* corresponds to the highlighted square?
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![[ordering-y-repetition-n.jpg]]
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Back: $(n)_k$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709305912355-->
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END%%
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%%ANKI
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Basic
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What combinatorial *concept* corresponds to the highlighted square?
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![[ordering-y-repetition-n.jpg]]
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Back: $k$-permutations (falling factorials)
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709306052449-->
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END%%
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%%ANKI
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Basic
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Which square corresponds to notation $(n)_k$?
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![[ordering-repetition.jpg]]
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Back:
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![[ordering-y-repetition-n.jpg]]
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<!--ID: 1709305912359-->
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END%%
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%%ANKI
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Basic
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What combinatorial *notation* corresponds to the highlighted square?
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![[ordering-n-repetition-y.jpg]]
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Back: $\binom{n + k}{k}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709306052455-->
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END%%
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%%ANKI
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Basic
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What combinatorial *concept* corresponds to the highlighted square?
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![[ordering-n-repetition-y.jpg]]
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Back: Stars and bars
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709306052461-->
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END%%
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%%ANKI
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Basic
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Which square corresponds to notation $\binom{n + k}{k}$?
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![[ordering-repetition.jpg]]
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Back:
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![[ordering-n-repetition-y.jpg]]
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<!--ID: 1709306052468-->
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END%%
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%%ANKI
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Basic
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What combinatorial *notation* corresponds to the highlighted square?
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![[ordering-n-repetition-n.jpg]]
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Back: $\binom{n}{k}$
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709306140856-->
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END%%
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%%ANKI
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Basic
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What combinatorial *concept* corresponds to the highlighted square?
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![[ordering-n-repetition-n.jpg]]
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Back: Combinations
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Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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<!--ID: 1709306140887-->
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END%%
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%%ANKI
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Basic
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Which square corresponds to notation $\binom{n}{k}$?
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![[ordering-repetition.jpg]]
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Back:
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![[ordering-n-repetition-n.jpg]]
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<!--ID: 1709306140891-->
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END%%
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## Bibliography
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* Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n.d., [https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf](https://discrete.openmathbooks.org/pdfs/dmoi3-tablet.pdf).
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