From 2678c94882fe61be6a369dcc175fd55c5098f5db Mon Sep 17 00:00:00 2001 From: Joshua Potter Date: Wed, 21 Feb 2024 10:51:48 -0700 Subject: [PATCH] Move hexadecimal to radices page. --- .../plugins/obsidian-to-anki-plugin/data.json | 17 +-- notes/_journal/2024-02-21.md | 11 ++ notes/_journal/{ => 2024-02}/2024-02-20.md | 0 notes/algebra/floor-ceiling.md | 12 +- .../hexadecimal.md => algebra/radices.md} | 111 +++++++++++++++++- notes/binary/index.md | 2 +- notes/c/escape-sequences.md | 2 +- notes/combinatorics/combinations.md | 72 +++++++++++- notes/combinatorics/permutations.md | 2 +- notes/posix/regexp.md | 2 +- 10 files changed, 208 insertions(+), 23 deletions(-) create mode 100644 notes/_journal/2024-02-21.md rename notes/_journal/{ => 2024-02}/2024-02-20.md (100%) rename notes/{binary/hexadecimal.md => algebra/radices.md} (62%) diff --git a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json index f18a3f0..9cb0222 100644 --- a/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json +++ b/notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json @@ -90,7 +90,7 @@ "bash/quoting.md": "b1d8869a91001f8b22f0cdc54d806f61", "bash/robustness.md": "7ab094b95ba2bfa885adba8e9efedf68", "bash/shebang.md": "9006547710f9a079a3666169fbeda7aa", - "c/escape-sequences.md": "0d6219ebb51f6f21e026de67603e25b8", + "c/escape-sequences.md": "7b4bbf159908320249158acfe47a9074", "c/index.md": "a021c92f19831bdd2bca4cbf813882fe", "gawk/index.md": "dd851e023e11c556c0272a0dcb6dd55d", "gawk/variables.md": "73b12bd0d7d6f97b4a7285aaf2c45bfa", @@ -107,7 +107,7 @@ "posix/index.md": "97b1b8ecb9a953e855a9acf0ab25b8c8", "posix/signals.md": "4fe63c3c9507b2e15c9ad6f3a2b541db", "templates/daily.md": "7866014e730e85683155207a02e367d8", - "posix/regexp.md": "43825a1b9ed0dd7eeb1b6fe35c928bfe", + "posix/regexp.md": "f5fb177c7356faf1bf768023c2563c54", "journal/2024-02-04.md": "e2b5678fc53d7284b71ed6820c02b954", "gawk/regexp.md": "d9229f1eabe1b99e965eecaa03bee86c", "_templates/daily.md": "7866014e730e85683155207a02e367d8", @@ -128,7 +128,7 @@ "algorithms/sorting/selection-sort.md": "fcd0dc2ebaabd0a4db1baf7e7ef9f7a9", "algorithms/index 1.md": "6fada1f3d5d3af64687719eb465a5b97", "binary/hexadecimal.md": "c3d485f1fd869fe600334ecbef7d5d70", - "binary/index.md": "ab345e75dc01f890faa31bc26676d526", + "binary/index.md": "d97bddf94227df5903a2929febf25606", "_journal/2024-02-09.md": "a798d35f0b2bd1da130f7ac766166109", "c/types.md": "cf3e66e5aee58a94db3fdf0783908555", "logic/quantification.md": "5d7579a511e9ff683edeec62bcc291b8", @@ -166,7 +166,7 @@ "_journal/2024-02/2024-02-14.md": "aa009f9569e175a8104b0537ebcc5520", "_journal/2024-02-16.md": "5cc129254afd553829be3364facd23db", "_journal/2024-02/2024-02-15.md": "16cb7563d404cb543719b7bb5037aeed", - "algebra/floor-ceiling.md": "456fa31bedb9ec7c2fa1d6f75db81dec", + "algebra/floor-ceiling.md": "efc4502ed22128e14b20ba88b368a872", "algebra/index.md": "90b842eb694938d87c7c68779a5cacd1", "algorithms/binary-search.md": "08cb6dc2dfb204a665d8e8333def20ca", "_journal/2024-02-17.md": "7c37cb10515ed3d2f5388eaf02a67048", @@ -179,11 +179,14 @@ "combinatorics/additive-principle.md": "84dcd0243263b3c53456086ae43fa00f", "_journal/2024-02-19.md": "30d16c5373deb9cb128d2e7934ae256a", "_journal/2024-02/2024-02-18.md": "67e36dbbb2cac699d4533b5a2eaeb629", - "combinatorics/permutations.md": "9351e4d5c4457c34198640cf04bdd888", - "combinatorics/combinations.md": "2e7069e018525e10e4e2b9fb46bc8291", + "combinatorics/permutations.md": "606b4b2b8018797ca54857112235d96e", + "combinatorics/combinations.md": "6fc179a9bf4e3958f28c4c3f7da5cda0", "_journal/2024-02-20.md": "b85ba0eeeb16e30a602ccefabcc9763e", "_journal/2024-02/2024-02-19.md": "df1a9ab7ab89244021b3003c84640c78", - "combinatorics/inclusion-exclusion.md": "4d5ba716bc90cd378c7c4c816b224c75" + "combinatorics/inclusion-exclusion.md": "4d5ba716bc90cd378c7c4c816b224c75", + "_journal/2024-02-21.md": "d7545ab01e651e565c056ded99455286", + "_journal/2024-02/2024-02-20.md": "af2ef10727726200c4defe2eafc7d841", + "algebra/radices.md": "1171835a864103fbc4cbe21d56fcfa1f" }, "fields_dict": { "Basic": [ diff --git a/notes/_journal/2024-02-21.md b/notes/_journal/2024-02-21.md new file mode 100644 index 0000000..2af5a74 --- /dev/null +++ b/notes/_journal/2024-02-21.md @@ -0,0 +1,11 @@ +--- +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) \ No newline at end of file diff --git a/notes/_journal/2024-02-20.md b/notes/_journal/2024-02/2024-02-20.md similarity index 100% rename from notes/_journal/2024-02-20.md rename to notes/_journal/2024-02/2024-02-20.md diff --git a/notes/algebra/floor-ceiling.md b/notes/algebra/floor-ceiling.md index 59172ba..7dfb5f7 100644 --- a/notes/algebra/floor-ceiling.md +++ b/notes/algebra/floor-ceiling.md @@ -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). 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). 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). @@ -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). @@ -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). 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). END%% diff --git a/notes/binary/hexadecimal.md b/notes/algebra/radices.md similarity index 62% rename from notes/binary/hexadecimal.md rename to notes/algebra/radices.md index ae79fbc..1ed56f5 100644 --- a/notes/binary/hexadecimal.md +++ b/notes/algebra/radices.md @@ -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). + +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). + +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). + +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). + +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 + +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 + +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 + +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 + +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 + +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 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 + 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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). \ No newline at end of file diff --git a/notes/binary/index.md b/notes/binary/index.md index 2651419..bb4ac34 100644 --- a/notes/binary/index.md +++ b/notes/binary/index.md @@ -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) { diff --git a/notes/c/escape-sequences.md b/notes/c/escape-sequences.md index 392b17d..c30712c 100644 --- a/notes/c/escape-sequences.md +++ b/notes/c/escape-sequences.md @@ -49,7 +49,7 @@ Tags: bash 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. diff --git a/notes/combinatorics/combinations.md b/notes/combinatorics/combinations.md index c2e160e..b8acbfe 100644 --- a/notes/combinatorics/combinations.md +++ b/notes/combinatorics/combinations.md @@ -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). 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). + +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 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). + +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). + +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). + +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). + +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). + +END%% + +%%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). + +END%% + +%%ANKI +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). + +END%% + ## 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]]. diff --git a/notes/combinatorics/permutations.md b/notes/combinatorics/permutations.md index 791b178..5d2384f 100644 --- a/notes/combinatorics/permutations.md +++ b/notes/combinatorics/permutations.md @@ -188,7 +188,7 @@ END%% %%ANKI 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). diff --git a/notes/posix/regexp.md b/notes/posix/regexp.md index bcbe1dc..cecd6a3 100644 --- a/notes/posix/regexp.md +++ b/notes/posix/regexp.md @@ -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 %%ANKI Basic