From c1a19058ef1b3fceed7d958705b3f01f27c208ae Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Fri, 23 Feb 2024 12:08:50 -0700 Subject: [PATCH] Combinatorics matrix on counting strategies. --- .../plugins/obsidian-to-anki-plugin/data.json | 6 +- notes/_journal/2024-02-23.md | 7 +- notes/combinatorics/index.md | 68 +++++++++++++++++++ notes/combinatorics/permutations.md | 2 + 4 files changed, 78 insertions(+), 5 deletions(-) diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index 9efb163..a41ce05 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -172,14 +172,14 @@ "_journal/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048", "_journal/2024-02/2024-02-16.md": "e701902e369ec53098fc2deed4ec14fd", "binary/integer-encoding.md": "ed39771ee9de86423f75fda74b8c5aa2", - "combinatorics/index.md": "f9de9671fdb6068ef2bb5e63051734be", + "combinatorics/index.md": "9a85e8858c50c9797243d6d01e1dcbe7", "_journal/2024-02-18.md": "67e36dbbb2cac699d4533b5a2eaeb629", "_journal/2024-02/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048", "combinatorics/multiplicative-principle.md": "eae60248d68ba8bd2da5e4c3fea70109", "combinatorics/additive-principle.md": "e968028670f95ee9a7c5499ff7cb6792", "_journal/2024-02-19.md": "30d16c5373deb9cb128d2e7934ae256a", "_journal/2024-02/2024-02-18.md": "67e36dbbb2cac699d4533b5a2eaeb629", - "combinatorics/permutations.md": "606b4b2b8018797ca54857112235d96e", + "combinatorics/permutations.md": "bde773a70e212e669ff4b8bae0f1d1f0", "combinatorics/combinations.md": "6fc179a9bf4e3958f28c4c3f7da5cda0", "_journal/2024-02-20.md": "b85ba0eeeb16e30a602ccefabcc9763e", "_journal/2024-02/2024-02-19.md": "df1a9ab7ab89244021b3003c84640c78", @@ -189,7 +189,7 @@ "algebra/radices.md": "0fcd901c798eaed8075ff1375e2429dd", "_journal/2024-02-22.md": "e01f1d4bd2f7ac2a667cdfd500885a2a", "_journal/2024-02/2024-02-21.md": "f423137ae550eb958378750d1f5e98c7", - "_journal/2024-02-23.md": "9750fc77ccd67fd29dc5af0cdc0d16b7", + "_journal/2024-02-23.md": "75eec3feffa90a219de77151771fe3fe", "_journal/2024-02/2024-02-22.md": "312e55d57868026f6e80f7989a889c2b", "c17/strings.md": "bbe8983602adbeb38eff214beddedd84", "c17/index.md": "78576ee41d0185df82c59999142f4edb", diff --git a/notes/_journal/2024-02-23.md b/notes/_journal/2024-02-23.md index 44f24c2..0bd53ec 100644 --- a/notes/_journal/2024-02-23.md +++ b/notes/_journal/2024-02-23.md @@ -8,7 +8,7 @@ title: "2024-02-23" - [x] OGS (1 Life & Death Problem) - [ ] Korean (Read 1 Story) - [ ] Interview Prep (1 Practice Problem) -- [ ] Log Work Hours (Max 3 hours) +- [x] Log Work Hours (Max 3 hours) * 101weiqi (serial numbers) * Q-28857 @@ -16,4 +16,7 @@ title: "2024-02-23" * Q-123426 * Q-10929 * Q-10924 - * Q-9107 \ No newline at end of file + * Q-9107 +* Read about extension and truncation of integral values using unsigned and two's-complement encoding. +* Read through last sections of "Discrete Mathematics: An Open Introduction"'s first chapter. + * I took a few little notes but I didn't pay as close attention to these sections as others. \ No newline at end of file diff --git a/notes/combinatorics/index.md b/notes/combinatorics/index.md index bfcb865..de5ec68 100644 --- a/notes/combinatorics/index.md +++ b/notes/combinatorics/index.md @@ -1,3 +1,71 @@ --- title: Combinatorics +TARGET DECK: Obsidian::STEM +FILE TAGS: combinatorics set +tags: + - combinatorics + - set --- + +## Overview + +When selecting objects, we can use the given table to hint at what counting strategy we should use: + +Order | Repeats | Answer Shape | Reference +----- | ------- | ------------------ | --------- +Yes | Yes | $n^k$ | `-` +Yes | No | $(n)_k$ | [[permutations#Falling Factorials]] +No | Yes | $\binom{n + k}{k}$ | [[combinations#Stars and Bars]] +No | No | $\binom{n}{k}$ | [[combinations]] + +%%ANKI +Basic +What does it mean for order to matter? +Back: We get different outcomes if the same objects are selected in different orders. +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). + +END%% + +%%ANKI +Basic +What does it mean for repeats to be allowed? +Back: The same object can be selected multiple times. +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). + +END%% + +%%ANKI +Basic +If order matters and repeats are allowed, the number of selections is usually formatted in what way? +Back: $n^k$ +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). + +END%% + +%%ANKI +Basic +If order matters and repeats are disallowed, the number of selections is usually formatted in what way? +Back: $(n)_k$ (falling factorial) +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). + +END%% + +%%ANKI +Basic +If order does not matter and repeats are allowed, the number of selections is usually formatted in what way? +Back: $\binom{n + k}{k}$ (stars and bars) +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). + +END%% + +%%ANKI +Basic +If order does not matter and repeats are disallowed, the number of selections is usually formatted in what way? +Back: $\binom{n}{k}$ (combinations) +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). + +END%% + +## References + +* 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). diff --git a/notes/combinatorics/permutations.md b/notes/combinatorics/permutations.md index 5d2384f..909d93a 100644 --- a/notes/combinatorics/permutations.md +++ b/notes/combinatorics/permutations.md @@ -174,6 +174,8 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n END%% +## Falling Factorials + If we generalize to choosing $k \leq n$ elements of $k$ objects, we can calculate the $k$-permutation of $n$. This is denoted as $(n)_k$, sometimes called the **falling factorial**. $$(n)_k = \frac{n!}{(n - k)!}$$ The derivation works by noting that we have $n - 0$ possible ways to pick the first object, $n - 1$ ways to pick the second, up until $n - (k - 1)$ ways to pick the last object.