Combinatorics matrix on counting strategies.

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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
+	* 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.

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).
+<!--ID: 1708715147778-->
+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).
+<!--ID: 1708715147781-->
+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).
+<!--ID: 1708715147783-->
+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).
+<!--ID: 1708715147784-->
+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).
+<!--ID: 1708715147786-->
+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).
+<!--ID: 1708715147787-->
+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).

notes/combinatorics/permutations.md

@@ -174,6 +174,8 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708366788613-->
 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.