More notes on combinations/binomial coefficients.

notes/.obsidian/plugins/obsidian-to-anki-plugin/data.json

@@ -75,7 +75,11 @@
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-    "bubble-sort.gif"
+    "bubble-sort.gif",
+    "lattice-path-chessboard.png",
+    "lattice-path-chessboard-colored.png",
+    "lattice-path-before-recurrence.png",
+    "lattice-path-after-recurrence.png"
   ],
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notes/_journal/2024-02-19.md

@@ -11,3 +11,4 @@ title: "2024-02-19"
 - [ ] Log Work Hours (Max 3 hours)
 
 * Start encoding notes on permutations and combinations, as well as their relationship to one another.
+* Continuing on `printf` fields. Added flashcards for `width` and `precision`.

notes/c/strings.md

@@ -46,6 +46,21 @@ END%%
 
 The syntax for the format placeholder is `%[flags][width][.precision][length]specifier`.
 
+%%ANKI
+Basic
+Which header file contains `printf` functionality?
+Back: `stdio.h`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441467-->
+END%%
+
+%%ANKI
+Cloze
+{1:`width`} field specifies a {2:minimum} number of characters while {2:`precision`} specifies a {1:maximum}.
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441472-->
+END%%
+
 Flag      | Description
 --------- | -----------
 `-`       | Left-aligns the output
@@ -60,6 +75,86 @@ Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia
 <!--ID: 1707918756812-->
 END%%
 
+%%ANKI
+Basic
+What is the output of `printf("%-2d abc", 100)`?
+Back: `100 abc`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441477-->
+END%%
+
+%%ANKI
+Basic
+What is the output of `printf("%-4d abc", 100)`?
+Back: `100  abc`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441481-->
+END%%
+
+%%ANKI
+Basic
+What is the output of `printf("%.4d abc", 100)`?
+Back: `0100 abc`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441486-->
+END%%
+
+%%ANKI
+Basic
+What is the output of `printf("%04d abc", 100)`?
+Back: `0100 abc`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441491-->
+END%%
+
+%%ANKI
+Basic
+What is the output of `printf("%.4s abc", "efg")`?
+Back: `efg abc`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441495-->
+END%%
+
+%%ANKI
+Basic
+What is the output of `printf("%.2s abc", "efg")`?
+Back: `ef abc`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441499-->
+END%%
+
+%%ANKI
+Basic
+What is the output of `printf("%.2f abc", 0.01234)`?
+Back: `0.01 abc`
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441503-->
+END%%
+
+%%ANKI
+Basic
+How does the precision field affect `"%s"` parameters?
+Back: Strings are truncated if longer than the specified precision.
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441507-->
+END%%
+
+%%ANKI
+Basic
+How does the precision field affect `"%f"` parameters?
+Back: Decimal values are truncated if longer than the specified precision.
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441511-->
+END%%
+
+%%ANKI
+Basic
+What does a negative width field value indicate?
+Back: This is actually a positive width with a `-` (left-align) flag.
+Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
+<!--ID: 1708384441515-->
+END%%
+
 %%ANKI
 Basic
 The `-` flag overrides what default behavior?
@@ -168,7 +263,7 @@ END%%
 
 %%ANKI
 Cloze
-The {`d` and `i`} format specifers output a {decimal `signed int`}.
+The {`d` and `i`} format specifers work on a {decimal `signed int`}.
 Reference: “Printf,” in *Wikipedia*, January 18, 2024, [https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962](https://en.wikipedia.org/w/index.php?title=Printf&oldid=1196716962).
 Tags: printf
 <!--ID: 1707852083040-->

notes/combinatorics/combinations.md

@@ -160,6 +160,180 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708368078746-->
 END%%
 
+## Pascal's Triangle
+
+A visual representation of the binomial coefficient's is in the form of Pascal's Triangle:
+```
+1
+1   1
+1   2   1
+1   3   3   1
+1   4   6   4   1
+1   5  10  10   5   1
+...
+```
+
+Terms are generated by adding the two terms above it, formalized via recurrence $$\binom{n}{k} = \binom{n - 1}{k} + \binom{n - 1}{k - 1}$$
+
+%%ANKI
+Basic
+What recurrence relation does $\binom{n}{k}$ satisfy?
+Back: $$\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$$
+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: 1708384441360-->
+END%%
+
+%%ANKI
+Basic
+What name is given to the following structure?
+```
+    1
+   1 1
+  1 2 1
+ 1 3 3 1
+1 4 6 4 1
+```
+Back: Pascal's Triangle.
+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: 1708384441393-->
+END%%
+
+%%ANKI
+Basic
+What recurrence relation is Pascal's Triangle based on?
+Back: $$\binom{n}{k} = \binom{n - 1}{k - 1} + \binom{n - 1}{k}$$
+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: 1708384441399-->
+END%%
+
+%%ANKI
+Basic
+What binomial coefficient is at the peak of Pascal's Triangle?
+Back: $\binom{0}{0}$
+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: 1708384441405-->
+END%%
+
+%%ANKI
+Basic
+What is the value of $\sum_{k=0}^n \binom{n}{k}$?
+Back: $2^n$
+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: 1708384441412-->
+END%%
+
+%%ANKI
+Basic
+What is the sum of Pascal's Triangle's $n$th row?
+Back: $2^n$
+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: 1708384441417-->
+END%%
+
+%%ANKI
+Basic
+Why does $\sum_{k=0}^n \binom{n}{k} = 2^n$?
+Back: The summands count the number of subsets of a set of cardinality $n$.
+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: 1708384441423-->
+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 "0", we see every lattice path has a corresponding bit string.
+
+![[lattice-path-example.png]]
+
+In this example, the total number of lattice paths from point $(0, 0)$ to $(3, 2)$ is therefore $\binom{5}{2} = \binom{5}{3}$.
+
+%%ANKI
+Basic
+What is the integer lattice?
+Back: $\{(x, y) : x \in \omega \land y \in \omega\}$
+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: 1708384441429-->
+END%%
+
+%%ANKI
+Basic
+What is a lattice path?
+Back: A possible shortest path between two points on an integer lattice.
+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: 1708384441434-->
+END%%
+
+%%ANKI
+Basic
+How many lattice paths exist from one corner of a chess board to the opposite corner?
+Back: $\binom{16}{8}$
+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: 1708384441440-->
+END%%
+
+%%ANKI
+Basic
+How many horizontal moves must a lattice path make between $(0, 0)$ and $(3, 2)$?
+Back: $3$
+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: 1708384507600-->
+END%%
+
+%%ANKI
+Basic
+How many vertical moves must a lattice path make between $(0, 0)$ and $(3, 2)$?
+Back: $2$
+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: 1708384507605-->
+END%%
+
+%%ANKI
+Basic
+How many moves (horizontal or vertical) must a lattice path make between $(0, 0)$ and $(3, 2)$?
+Back: $5$
+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: 1708384507608-->
+END%%
+
+%%ANKI
+Basic
+How do lattice paths relate to bit strings?
+Back: Denoting horizontal and vertical moves by $H$ and $V$ makes the relationship clear.
+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: 1708384507611-->
+END%%
+
+%%ANKI
+Basic
+How many lattice paths exist between $(0, 0)$ and $(3, 2)$?
+Back: $\binom{5}{2} = \binom{5}{3}$
+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: 1708384507615-->
+END%%
+
+%%ANKI
+Basic
+How is the number of lattice paths in the following shown to mirror the binomial recurrence?
+![[lattice-path-before-recurrence.png]]
+Back:
+It equals the number of lattice paths to $A$ plus the number of lattice paths to $B$.
+![[lattice-path-after-recurrence.png]]
+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: 1708384507618-->
+END%%
+
+%%ANKI
+Basic
+The number of lattice paths to the marked square can be derived by adding the values of which two squares?
+![[lattice-path-chessboard.png]]
+Back:
+![[lattice-path-chessboard-colored.png]]
+<!--ID: 1708384441446-->
+END%%
+
+## Binomials
+
+A **binomial** is a polynomial containing two terms. Consider $(x + y)^n$.
+
 %%ANKI
 Basic
 What term describes e.g. $\binom{n}{1}$, $\binom{n}{2}$, etc.?
@@ -178,12 +352,36 @@ END%%
 
 %%ANKI
 Basic
-What are binomial coefficients?
+Why are binomial coefficients named the way they are?
-The coefficients of terms in the expansion of a binomial, e.g. $(x + y)^n$.
+Back: They are the coefficients of terms in the expansion of a binomial, e.g. $(x + y)^n$.
 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: 1708368078764-->
 END%%
 
+%%ANKI
+Basic
+How does $(x + y)^n$ relate to bit strings?
+Back: Each term $x^ky^{n-k}$ corresponds to a bit string with $k$ `1`s and $n - k$ `0`s.
+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: 1708384441451-->
+END%%
+
+%%ANKI
+Basic
+What is the coefficient of $x^3y^{12}$ in $(x + y)^{15}$?
+Back: $\binom{15}{3} = \binom{15}{12}$.
+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: 1708384441456-->
+END%%
+
+%%ANKI
+Basic
+How is $(x + y)^n$ written to mirror the binomial recurrence?
+Back: $(x + y)^n = x \cdot (x + y)^{n-1} + y \cdot (x + y)^{n - 1}$
+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: 1708384441462-->
+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).