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Move hexadecimal to radices page.

c-declarations
Joshua Potter 2024-02-21 10:51:48 -07:00
parent 886fb34762
commit 2678c94882
10 changed files with 208 additions and 23 deletions
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17
notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json
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@ -90,7 +90,7 @@
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@ -179,11 +179,14 @@
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notes/_journal/2024-02-21.md Normal file
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---
title: "2024-02-21"
---
- [x] Anki Flashcards
- [x] KoL
- [ ] Sheet Music (10 min.)
- [ ] OGS (1 Life & Death Problem)
- [ ] Korean (Read 1 Story)
- [ ] Interview Prep (1 Practice Problem)
- [ ] Log Work Hours (Max 3 hours)

0
notes/_journal/2024-02-20.md → notes/_journal/2024-02/2024-02-20.md
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12
notes/algebra/floor-ceiling.md
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@ -21,7 +21,7 @@ END%%
%%ANKI
Basic
What is the floor of $x$?
Back: The greatest integer less than $x$.
Back: The greatest integer less than or equal to $x$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779649-->
END%%
@ -37,7 +37,7 @@ END%%
%%ANKI
Basic
What is the ceiling of $x$?
Back: The least integer greater than $x$.
Back: The least integer greater than or equal to $x$.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779663-->
END%%
@ -68,7 +68,7 @@ END%%
%%ANKI
Basic
What can be said about $x$ if $\lceil x \rceil - \lfloor x \rceil = 0$?
What can be said about $x$ if $\lceil x \rceil - \lfloor x \rfloor = 0$?
Back: $x$ is even.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779687-->
@ -76,7 +76,7 @@ END%%
%%ANKI
Basic
What can be said about $x$ if $\lceil x \rceil - \lfloor x \rceil = 1$?
What can be said about $x$ if $\lceil x \rceil - \lfloor x \rfloor = 1$?
Back: $x$ is odd.
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708110779693-->
@ -195,7 +195,7 @@ END%%
%%ANKI
Basic
If `A[p..q]` has odd size, what `r` most fairly allows partitions `A[p..r]` and `A[r+1..q]`?
Back: $r = \lfloor (p + q) / 2 \rfloor$
Back: $r = \lfloor (p + q) / 2 \rfloor = \lceil (p + q) / 2 \rceil$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708114757958-->
END%%
@ -203,7 +203,7 @@ END%%
%%ANKI
Basic
If `A[p..q]` has odd size, what `r` most fairly allows partitions `A[p..r-1]` and `A[r..q]`?
Back: $r = \lceil (p + q) / 2 \rceil$
Back: $r = \lfloor (p + q) / 2 \rfloor = \lceil (p + q) / 2 \rceil$
Reference: Ronald L. Graham, Donald Ervin Knuth, and Oren Patashnik, *Concrete Mathematics: A Foundation for Computer Science*, 2nd ed (Reading, Mass: Addison-Wesley, 1994).
<!--ID: 1708114757961-->
END%%

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notes/binary/hexadecimal.md → notes/algebra/radices.md
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@ -1,26 +1,108 @@
---
title: Hexadecimal
title: Radices
TARGET DECK: Obsidian::STEM
FILE TAGS: binary::hex
FILE TAGS: algebra
tags:
- binary
- hexadecimal
- algebra
---
## Overview
Hexadecimal encoding refers to the 16-base representation of binary numbers. Distinguish potentially ambiguous values like $32$ with the base as a subscript, e.g. $32_{10}$ vs $32_{16}$.
The **radix** is the number of unique digits used to represent numbers in a positional numeral system. Most commonly used systems tend to be binary ($2$-base), octal ($8$-base), decimal ($10$-base), and [[#Hexadecimal|hexadecimal]] ($16$-base).
%%ANKI
Basic
What is the process of subtracting a larger digit from a smaller one in radix $r$?
Back: Decrement the next non-zero and add $r$ to the smaller digit in question.
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
<!--ID: 1708534662981-->
END%%
%%ANKI
Basic
What does the first step in the subtraction process of $100_2 - 10_2$ *look* like?
Back: $020_2 - 10_2$
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
<!--ID: 1708534662989-->
END%%
%%ANKI
Basic
In a positional numeral system, what does "radix" refer to?
Back: The number of unique digits used to represent numbers.
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
<!--ID: 1708534662993-->
END%%
%%ANKI
Basic
What is the radix of the decimal system?
Back: $10$
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
<!--ID: 1708534662997-->
END%%
%%ANKI
Basic
What is the radix of the octal system?
Back: $8$
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
Tags: binary
<!--ID: 1708534663001-->
END%%
%%ANKI
Basic
What is the radix of the hexadecimal system?
Back: $16$
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
Tags: binary::hex
<!--ID: 1708534663005-->
END%%
%%ANKI
Basic
What is the radix of the binary system?
Back: $2$
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
Tags: binary
<!--ID: 1708534663009-->
END%%
## Hexadecimal
Hexadecimal is a 16-base numeral system, usually represented with digits `0` to `9` and `a` to `f` or `A` to `F`.
%%ANKI
Cloze
A hexadecimal digit represents {4} bits.
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
Tags: binary::hex
<!--ID: 1708534663013-->
END%%
%%ANKI
Cloze
An octal digit represents {3} bits.
Reference: “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).
Tags: binary
<!--ID: 1708534663018-->
END%%
%%ANKI
Cloze
A byte consists of {2} hexadecimal digits.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641563-->
END%%
%%ANKI
Cloze
A nibble consists of {1} hexadecimal digits.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1708534663022-->
END%%
%%ANKI
@ -28,6 +110,7 @@ Basic
Hexadecimal digits are represented by what characters?
Back: `a` to `f`, `A` to `F`, and `0` to `9`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641565-->
END%%
@ -36,6 +119,7 @@ Basic
How does C denote a hexadecimal numeric constant?
Back: With `0x` or `0X`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex c
<!--ID: 1707432641567-->
END%%
@ -44,6 +128,7 @@ Basic
What is the decimal equivalent of hex `A`, `C`, and `F`?
Back: `10`, `12`, and `15` respectively.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641568-->
END%%
@ -52,6 +137,7 @@ Basic
What is the hexadecimal equivalent of decimal `11`, `12`, and `14`?
Back: `B`, `C`, and `E` respectively.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641570-->
END%%
@ -60,6 +146,7 @@ Basic
*When* should padding be introduced in binary to hexadecimal conversion?
Back: When the number of bits is not a multiple of `4`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641571-->
END%%
@ -68,6 +155,7 @@ Basic
*Where* is padding introduced in binary to hexadecimal conversion?
Back: To the left of the binary sequence.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641573-->
END%%
@ -76,6 +164,7 @@ Basic
What are the possible hex values the first digit of $2^n$ can take on?
Back: `1`, `2`, `4`, and `8`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641579-->
END%%
@ -84,6 +173,7 @@ Basic
What are the possible values in binary that the first nibble of $2^n$ can take on?
Back: `0001`, `0010`, `0100`, and `1000`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641580-->
END%%
@ -92,6 +182,7 @@ Basic
How is $j$ interpreted in the hex representation of $2^{i + 4j}$?
Back: As the number of `0`s in the encoding.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641582-->
END%%
@ -100,6 +191,7 @@ Basic
How is the $0$ in $2^{0 + 4j}$ translated to hex?
Back: As hex digit `1`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641583-->
END%%
@ -108,6 +200,7 @@ Basic
How is the $1$ in $2^{1 + 4j}$ translated to hex?
Back: As hex digit `2`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641585-->
END%%
@ -116,6 +209,7 @@ Basic
How is the $2$ (power) in $2^{2 + 4j}$ translated to hex?
Back: As hex digit `4`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641586-->
END%%
@ -124,6 +218,7 @@ Basic
How is the $3$ in $2^{3 + 4j}$ translated to hex?
Back: As hex digit `8`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641587-->
END%%
@ -132,6 +227,7 @@ Basic
How is $n$ in $2^n$ factored to quickly write the decimal value's hex representation?
Back: $n = i + 4j$ where $0 \leq i \leq 3$.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641589-->
END%%
@ -140,6 +236,7 @@ Basic
How is the *remainder* of e.g. `158 / 16` managed in decimal to hex conversion?
Back: As the next least significant bit of our conversion.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641594-->
END%%
@ -148,6 +245,7 @@ Basic
How is the *quotient* of e.g. `158 / 16` managed in decimal to hex conversion?
Back: As the next value to divide by `16`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641595-->
END%%
@ -156,6 +254,7 @@ Basic
When does repeated division in decimal to hex conversion end?
Back: When the quotient (*not* the remainder) is `0`.
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707919792632-->
END%%
@ -164,9 +263,11 @@ Basic
How is e.g. `0xAC32` expressed as a sum of decimal values?
Back: $(16^3 \times 10) + (16^2 \times 12) + (16^1 \times 3) + (16^0 \times 2)$
Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
Tags: binary::hex
<!--ID: 1707432641596-->
END%%
## References
* Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
* “Radix,” in *Wikipedia*, August 6, 2023, [https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173](https://en.wikipedia.org/w/index.php?title=Radix&oldid=1169046173).

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notes/binary/index.md
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@ -8,7 +8,7 @@ tags:
## Overview
A binary digit or **bit** is a `0` or `1` character. A **bit string** is then a contiguous sequence of bits. It's **weight** is a reference to the number of `1`s in the bit string. Compare the below operation to the method for converting from one numerical base to another (e.g. [[hexadecimal]]).
A binary digit or **bit** is a `0` or `1` character. A **bit string** is then a contiguous sequence of bits. It's **weight** is a reference to the number of `1`s in the bit string. Compare the below operation to the method for converting from one numerical base to another (e.g. [[radices#Hexadecimal|hexadecimal]]).
```c
unsigned int bit_weight(int64_t n) {

2
notes/c/escape-sequences.md
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@ -49,7 +49,7 @@ Tags: bash
<!--ID: 1706975891817-->
END%%
* `\xhh`: Consists of one or more hexadecimal digits. The `x` prefix is required to distinguish from octal escape sequences.
* `\xhh`: Consists of one or more [[radices#Hexadecimal|hexadecimal]] digits. The `x` prefix is required to distinguish from octal escape sequences.
* [[bash/index|Bash]] supports this sequence as `$'\xhh'`. One or two digits is supported.
* [[gawk/index|gawk]] limits processing to two digits.
* Robbins states that using more than two hexadecimal digits can produce undefined results.

72
notes/combinatorics/combinations.md
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@ -90,12 +90,20 @@ END%%
%%ANKI
Basic
How is the closed formula of $\binom{n}{k}$ written in terms of factorials (*not* falling factorials)?
Without using falling factorials, what is the closed formula of $\binom{n}{k}$?
Back: $$\binom{n}{k} = \frac{n!}{k!(n - k)!}$$
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: 1708368078700-->
END%%
%%ANKI
Basic
Using falling factorials, what is the closed formula of $\binom{n}{k}$?
Back: $$\binom{n}{k} = \frac{(n)_k}{k!}$$
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: 1708532069694-->
END%%
%%ANKI
Basic
How do $k$-permutations of $n$ objects relate to $k$-combinations?
@ -354,6 +362,68 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
<!--ID: 1708434662166-->
END%%
### Stars and Bars
The **stars and bars** chart refers to a graphical depiction of distributing $n$ objects (represented as $*$) into $m$ different buckets (delineated via $|$. An example chart looks like so: $$**|***|*||*$$
Notice there are $m - 1$ bars and interspersed amongst the $n$ stars. In the above example, there are $11$ total symbols, $4$ of which are bars, meaning there are $\binom{11}{4}$ ways to distribute the objects amongst the $5$ buckets. We can represent this using bit strings instead, with `0`s as stars and `1`s as bars. The above example is equivalently written as: $$00100010110$$
%%ANKI
Basic
What symbols are typically used in a stars and bars chart?
Back: $*$ and $|$
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: 1708532069700-->
END%%
%%ANKI
Basic
What property is exhibited by objects distributed in a stars and bars chart?
Back: They are identical to one another.
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: 1708532069704-->
END%%
%%ANKI
Basic
What kind of chart is the following an example of? $$**|***|*||*$$
Back: 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: 1708532069707-->
END%%
%%ANKI
Basic
How is the following stars and bars chart written as a bit string? $$**|***|*||*$$
Back: $00100010110$
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: 1708532069711-->
END%%
%%ANKI
Basic
The following stars and bars chart is a single instance of how many possible choices? $$**|***|*||*$$
Back: $\binom{11}{4}$
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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%%ANKI
Basic
What graphical depiction is used to find the number of integer solutions to the following equation? $$x_1 + x_2 + \cdots + x_k = n$$
Back: 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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Basic
What *is* the distribution corresponding to the following stars and bars chart? $$*||*$$
Back: A single object in the first and last bucket. No object in the middle.
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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## Lattice Paths
A **lattice path** is one of the shorted possible paths connecting two points on a lattice, moving only horizontally and vertically. By representing each horizontal move by `1` and each vertical move by `1`, we see every lattice path has a corresponding [[#Bit Strings|bit string]].

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Basic
What is the closed formula for $(n)_k$ (falling factorial)?
What is the closed formula for falling factorial $(n)_k$?
Back: $$(n)_k = \frac{n!}{(n - k)!}$$
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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@ -294,7 +294,7 @@ Class | Similar To | Meaning
`[:punct:]` | | All graphic characters except letters and digits
`[:space:]` | `[ \t\n\r\f\v]` | Whitespace characters
`[:upper:]` | `[A-Z]` | Uppercase alphabetic characters
`[:xdigit:]` | `[0-9A-Fa-f]` | Hexadecimal digits
`[:xdigit:]` | `[0-9A-Fa-f]` | [[radices#Hexadecimal\|Hexadecimal]] digits
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Basic