Combinatorics matrix on counting strategies.

c-declarations
Joshua Potter 2024-02-23 12:08:50 -07:00
parent 15001806d0
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- [x] OGS (1 Life & Death Problem) - [x] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story) - [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem) - [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours) - [x] Log Work Hours (Max 3 hours)
* 101weiqi (serial numbers) * 101weiqi (serial numbers)
* Q-28857 * Q-28857
@ -17,3 +17,6 @@ title: "2024-02-23"
* Q-10929 * Q-10929
* Q-10924 * 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.

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--- ---
title: Combinatorics 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).
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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).
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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).
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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).
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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).
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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).
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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).

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END%% 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**. $$(n)_k = \frac{n!}{(n - k)!}$$ 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. 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.