Algorithmic notation and more on two's-complement.

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

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notes/_journal/2024-02-26.md

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----
-title: "2024-02-26"
----
-
-- [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)
-
-* Added flashcards on two's-complement addition.
-* Consolidated `integer-encoding.md` with `shifts.md`.

notes/_journal/2024-02-27.md

@@ -0,0 +1,11 @@
+---
+title: "2024-02-27"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [x] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [ ] Interview Prep (1 Practice Problem)
+- [ ] Log Work Hours (Max 3 hours)

notes/_journal/2024-02/2024-02-26.md

@@ -0,0 +1,29 @@
+---
+title: "2024-02-26"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [x] OGS (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [x] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)
+
+* Added flashcards on two's-complement addition.
+* Consolidated `integer-encoding.md` with `shifts.md`.
+* Added notes on $\Theta$-notation.
+* Read chapter 2 of "Designing Data-Intensive Applications".
+* 101weiqi (serial numbers)
+	* Q-265171
+	* Q-40523
+	* Q-267801
+	* Q-368844
+	* Q-61135
+	* Q-12528
+	* Q-122965
+	* Q-77762
+* Leetcode
+	* [Longest Common Prefix](https://leetcode.com/problems/longest-common-prefix/)
+	* [Swap Nodes in Pairs](https://leetcode.com/problems/swap-nodes-in-pairs/)
+	* [Remove Duplicates From Sorted Array](https://leetcode.com/problems/remove-duplicates-from-sorted-array/)

notes/_templates/daily.md

@@ -5,7 +5,7 @@ title: "{{date:YYYY-MM-DD}}"
 - [ ] Anki Flashcards
 - [ ] KoL
 - [ ] Sheet Music (10 min.)
-- [ ] OGS (1 Life & Death Problem)
+- [ ] Go (1 Life & Death Problem)
 - [ ] Korean (Read 1 Story)
 - [ ] Interview Prep (1 Practice Problem)
 - [ ] Log Work Hours (Max 3 hours)

notes/algorithms/order-growth.md

@@ -91,6 +91,258 @@ Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambri
 <!--ID: 1707344177515-->
 END%%
 
+%%ANKI
+Basic
+How do we simplify $\Theta(an^2 + bn + c)$?
+Back: As $\Theta(n^2)$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221765-->
+END%%
+
+%%ANKI
+Basic
+Explain why asymptotic notation is useful for *both* running times and space usage.
+Back: Asymptotic notation represents functions in a general sense.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221769-->
+END%%
+
+%%ANKI
+Basic
+*Which* running time are algorithm designers typically concerned with?
+Back: Worst-case running time.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221774-->
+END%%
+
+%%ANKI
+Basic
+In asymptotic notation, how is constant space usage denoted?
+Back: Space usage is $O(1)$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221778-->
+END%%
+
+%%ANKI
+Basic
+How could we replace equality $f(n) = \Theta(g(n))$ to be less "abusive"?
+Back: Replace $=$ with $\in$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221783-->
+END%%
+
+%%ANKI
+Basic
+How is equality abused in $f(n) = \Theta(g(n))$?
+Back: Here $=$ actually refers to set membership.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221788-->
+END%%
+
+%%ANKI
+Basic
+How could we replace $1$ in $\Theta(1)$ to be less "abusive"?
+Back: Replace $1$ with $n^0$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221793-->
+END%%
+
+%%ANKI
+Basic
+*Why* does Cormen et al. consider $\Theta(1)$ to be a minor abuse?
+Back: This expression does not indicate what variable is tending to infinity.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221797-->
+END%%
+
+## $\Theta$-notation
+
+![[theta-notation.png]]
+
+$\Theta$-notation refers to a strict lower- and upper-bound. It is defined as set $$\Theta(g(n)) = \{ f(n) \mid \exists c_1, c_2, n_0 > 0, \forall n \geq n_0, 0 \leq c_1g(x) \leq f(n) \leq c_2g(n) \}$$
+
+%%ANKI
+Basic
+What kind of mathematical object is $\Theta(n)$?
+Back: A set.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221801-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers found in $\Theta(g(n))$, what values do $c_1$, $c_2$, and $n_0$ take on?
+Back: Positive constants.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221806-->
+END%%
+
+%%ANKI
+Basic
+What names are usually given to the existentially quantified identifers in $\Theta(g(n))$'s definition?
+Back: $c_1$, $c_2$, and $n_0$.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221811-->
+END%%
+
+%%ANKI
+Basic
+What name is usually given to the universally quantified identifer in $\Theta(g(n))$'s definition?
+Back: $n$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221815-->
+END%%
+
+%%ANKI
+Cloze
+Using typical identifiers, $f(n) = \Theta(g(n))$ satisfies {$0$} $\leq$ {$c_1g(n)$} $\leq$ {$f(n)$} $\leq$ {$c_2g(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221818-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the lower bound of $c_1g(n)$ in $\Theta(g(n))$?
+Back: $0$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221822-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $c_1g(n)$ in the definition of $\Theta(g(n))$?
+Back: $f(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221826-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the lower bound of $f(n)$ in the definition of $\Theta(g(n))$?
+Back: $c_1g(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221830-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $f(n)$ in the definition of $\Theta(g(n))$?
+Back: $c_2g(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221834-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the lower bound of $c_2g(n)$ in $\Theta(g(n))$?
+Back: $f(n)$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221839-->
+END%%
+
+%%ANKI
+Basic
+Using typical identifiers, what is the upper bound of $c_2g(n)$ in $\Theta(g(n))$?
+Back: N/A
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221844-->
+END%%
+
+%%ANKI
+Cloze
+Given $f(n) = \Theta(g(n))$, we say {1:$g(n)$} is an asymptotically {2:tight} bound for {1:$f(n)$}.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221851-->
+END%%
+
+%%ANKI
+Basic
+Which notation corresponds to asymptotically tight bounds?
+Back: $\Theta$-notation.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221857-->
+END%%
+
+%%ANKI
+Basic
+Every member of $\Theta(g(n))$ is expected to be asymptotically what?
+Back: Nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221864-->
+END%%
+
+%%ANKI
+What does it mean for function $f(n)$ to be asymptotically nonnegative?
+Back: $f(n) \geq 0$ whenever $n$ is sufficiently large.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+END%%
+
+%%ANKI
+Basic
+What does it mean for function $f(n)$ to be asymptotically positive?
+Back: $f(n) > 0$ whenever $n$ is sufficiently large.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221871-->
+END%%
+
+%%ANKI
+Basic
+What condition must $g(n)$ satisfy such that otherwise $\Theta(g(n))$ is empty?
+Back: $g(n)$ must be asymptotically nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221876-->
+END%%
+
+%%ANKI
+Basic
+What does $\Theta(-n)$ evaluate to?
+Back: $\varnothing$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221881-->
+END%%
+
+%%ANKI
+Basic
+*Why* is it $\Theta(-n) = \varnothing$?
+Back: Because $-n$ is not asymptotically nonnegative.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221886-->
+END%%
+
+%%ANKI
+Basic
+How is $\Theta(g(n))$ defined?
+Back: $\{ \exists c_1, c_2, n_0 > 0, \forall n \geq n_0, 0 \leq c_1g(n) \leq f(n) \leq c_2g(n) \}$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221892-->
+END%%
+
+%%ANKI
+Basic
+Using the typical identifiers, what values of $n$ are in the matrix of $\Theta(g(n))$'s definition?
+Back: $n \geq n_0$
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221898-->
+END%%
+
+%%ANKI
+Basic
+Which asymptotic notation is this image demonstrating?
+![[theta-notation.png]]
+Back: $\Theta$-notation
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221904-->
+END%%
+
+%%ANKI
+Basic
+What values does the $y$-axis implicitly range over in the following?
+![[theta-notation.png]]
+Back: Nonnegative values.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221909-->
+END%%
+
 ## References
 
 * Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).

notes/binary/integer-encoding.md

@@ -822,7 +822,395 @@ Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Program
 <!--ID: 1708701087982-->
 END%%
 
-## Shifting
+## Arithmetic
+
+### Addition
+
+Addition of two unsigned or two two's-complement numbers operate in much the same way as grade-school arithmetic. Digits are added one-by-one and overflows "carried" to the next summation. Overflows are truncated; the final carry bit is discarded in the underlying bit adder.
+
+%%ANKI
+Basic
+*Why* is adding $w$-bit integral types equal to $w$-bit truncation?
+Back: The underlying bit adder discards any final carry bit.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678721-->
+END%%
+
+%%ANKI
+Basic
+Why should you generally prefer `x < y` over `x - y < 0`?
+Back: The former avoids possible underflows.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678725-->
+END%%
+
+%%ANKI
+Basic
+How is `x - y < 0` rewritten more safely?
+Back: `x < y`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678728-->
+END%%
+
+%%ANKI
+Basic
+What hardware-level advantage does two's-complement introduce over other signed encodings?
+Back: The same circuits can be used for unsigned and two's-complement addition.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678732-->
+END%%
+
+%%ANKI
+Basic
+What representational-level advantage does two's-complement introduce over other signed encodings?
+Back: `0` is encoded in only one way.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678736-->
+END%%
+
+Unsigned addition of $w$-bit integral types, denoted $+_w^u$, behaves like so:
+
+$$x +_w^u y = \begin{cases}
+x + y - 2^w & \text{if } x + y \geq 2^w \\
+x + y & \text{otherwise}
+\end{cases}$$
+
+This is more simply expressed as $x +_w^u y = (x + y) \bmod 2^w$.
+
+%%ANKI
+Basic
+What kind of overflows does unsigned addition potentially exhibit?
+Back: Positive overflow.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678739-->
+END%%
+
+%%ANKI
+Basic
+Why is unsigned addition overflow *not* UB?
+Back: Because the C standard enforces unsigned encoding of `unsigned` data types.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1708799678742-->
+END%%
+
+%%ANKI
+Basic
+What does $+_w^u$ denote?
+Back: Unsigned addition of $w$-bit integral types.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678745-->
+END%%
+
+%%ANKI
+Basic
+Unsigned addition overflow is equivalent to what bit-level manipulation tactic?
+Back: Truncation.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678748-->
+END%%
+
+%%ANKI
+Basic
+What is the result of $x +_w^u y$?
+Back: $(x + y) \bmod 2^w$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678751-->
+END%%
+
+%%ANKI
+Basic
+*Why* does $x +_w^u y = (x + y) \bmod 2^w$?
+Back: Because discarding any carry bit is equivalent to truncating the sum to $w$ bits.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678755-->
+END%%
+
+%%ANKI
+Cloze
+Without using modular arithmetic, $x +_w^u y =$ {$x + y$} if {$x + y < 2^w$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678758-->
+END%%
+
+%%ANKI
+Cloze
+Without using modular arithmetic, $x +_w^u y =$ {$x + y - 2^w$} if {$x + y \geq 2^w$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678761-->
+END%%
+
+%%ANKI
+Basic
+How do you detect whether unsigned addition $s \coloneqq x +_w^u y$ overflowed?
+Back: Overflow occurs if and only if $s < x$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678765-->
+END%%
+
+%%ANKI
+Basic
+How would you complete the body of this function?
+```c
+/* Determine whether arguments can be added without overflow */
+int uadd_ok(unsigned x, unsigned y);
+```
+Back:
+```c
+return (x + y) >= x;
+```
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678769-->
+END%%
+
+%%ANKI
+Basic
+Does unsigned overflow detection depend on the left or right operand of $s \coloneqq x +_w^u y$?
+Back: Either.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678772-->
+END%%
+
+%%ANKI
+Basic
+Why can we compare $s$ to $x$ or $y$ when detecting overflow of $s \coloneqq x +_w^u y$?
+Back: Because unsigned addition is commutative.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708799678776-->
+END%%
+
+%%ANKI
+Basic
+Given integer $0 < x < 2^w$, what is $x$'s unsigned additive inverse?
+Back: $2^w - x$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708808252010-->
+END%%
+
+%%ANKI
+Basic
+Which unsigned integer is its own additive inverse?
+Back: $0$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708808252017-->
+END%%
+
+%%ANKI
+Basic
+What bitwise operations yield the additive inverse of an unsigned number $x$?
+Back: `~x + 1`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784783-->
+END%%
+
+%%ANKI
+Basic
+Given unsigned integer `x`, what is the value of `x + ~x`?
+Back: $UMax$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784788-->
+END%%
+
+Two's-complement addition, denoted $+_w^t$ operates similarly:
+
+$$x +_w^u y = \begin{cases}
+x + y - 2^w & \text{if } x + y \geq 2^{w-1} \\
+x + y + 2^w & \text{if } x + y < -2^{w-1} \\
+x + y & \text{otherwise}
+\end{cases}$$
+
+Unlike with unsigned addition, there is no simpler modulus operation that can be applied.
+
+%%ANKI
+Basic
+What kind of overflows does two's-complement addition potentially exhibit?
+Back: Positive and negative overflow.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376220-->
+END%%
+
+%%ANKI
+Basic
+Why is two's-complement addition overflow UB?
+Back: Because the C standard does not mandate any particular signed integer encoding.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+Tags: c17
+<!--ID: 1708964376225-->
+END%%
+
+%%ANKI
+Basic
+What does $+_w^t$ denote?
+Back: Two's-complement addition of $w$-bit integral types.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376228-->
+END%%
+
+%%ANKI
+Basic
+*Why* doesn't two's-complement addition perform modular arithmetic?
+Back: Because negative values are representable.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376231-->
+END%%
+
+%%ANKI
+Cloze
+$x +_w^t y =$ {$x + y - 2^w$} if {$x + y \geq 2^{w-1}$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376235-->
+END%%
+
+%%ANKI
+Cloze
+$x +_w^t y =$ {$x + y + 2^w$} if {$x + y < -2^{w-1}$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376238-->
+END%%
+
+%%ANKI
+Cloze
+$x +_w^t y =$ {$x + y$} if {$-2^{w-1} \leq x + y < 2^{w-1}$}.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376242-->
+END%%
+
+%%ANKI
+Basic
+How do we detect $x +_w^t y$ positive overflowed?
+Back: This happens iff $x > 0$, $y > 0$, and $x +_w^t y \leq 0$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376246-->
+END%%
+
+%%ANKI
+Basic
+How do we detect $x +_w^t y$ negative overflowed?
+Back: This happens iff $x < 0$, $y < 0$, and $x +_w^t y \geq 0$.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376250-->
+END%%
+
+%%ANKI
+Basic
+How can we write $x +_w^t y$ in terms of unsigned addition?
+Back: $x +_w^t y = U2T_w(T2U_w(x) +_w^u T2U_w(y))$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376254-->
+END%%
+
+%%ANKI
+*Why* are we able to characterize $+_w^t$ in terms of $+_w^u$?
+Back: Because two's-complement addition has the same bit-level representation as unsigned addition.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+END%%
+
+%%ANKI
+Basic
+How would you complete the body of this function?
+```c
+/* Determine whether arguments can be added without overflow */
+int tadd_ok(int x, int y);
+```
+Back:
+```c
+int pos_over = x > 0 && y > 0 && (x + y) <= 0;
+int neg_over = x < 0 && y < 0 && (x + y) >= 0;
+return !pos_over && !neg_over;
+```
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1708964376259-->
+END%%
+
+%%ANKI
+Basic
+Given integer $-2^{w-1} < x < 2^{w-1}$, what is $x$'s two's-complement additive inverse?
+Back: $-x$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709040965774-->
+END%%
+
+%%ANKI
+Basic
+What is the additive inverse of $TMin$?
+Back: $TMin$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709040965804-->
+END%%
+
+%%ANKI
+Basic
+What is the additive inverse of $TMax$?
+Back: $-TMax$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709040965810-->
+END%%
+
+%%ANKI
+Basic
+Which two's-complement integer is its own additive inverse?
+Back: $TMin$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709040965815-->
+END%%
+
+%%ANKI
+Basic
+What bitwise operations yield the additive inverse of two's-complement number $x$?
+Back: `~x + 1`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784791-->
+END%%
+
+%%ANKI
+Basic
+Given two's-complement integer `x`, what is the value of `x + ~x`?
+Back: $-1$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784794-->
+END%%
+
+%%ANKI
+Basic
+What "splitting" approach to $x$'s two's-complement negation does Bryant et al. describe?
+Back: Find the rightmost $1$ in $x$'s bit string representation and complement the bits to its left.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784797-->
+END%%
+
+%%ANKI
+Basic
+Where do we "split" $x$'s binary representation to perform two's-complement negation?
+Back: At the rightmost $1$ in $x$'s binary representation.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784800-->
+END%%
+
+%%ANKI
+Basic
+Using *just* `~`, what is the two's-complement negation of $\langle x_{w-1}, \ldots, x_{k+1}, 1, 0, \ldots, 0\rangle$?
+Back: $\langle \textasciitilde x_{w-1}, \ldots, \textasciitilde x_{k+1}, 1, 0, \ldots, 0 \rangle$
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784803-->
+END%%
+
+%%ANKI
+Basic
+*Why* does complementing and adding one yield integer $x$'s additive inverse?
+Back: `x + ~x` yields a bit string of all `1`s. Adding `1` to this overflows.
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709042784806-->
+END%%
+
+%%ANKI
+Basic
+What decimal value does two's-complement `~x` evaluate to?
+Back: `-x - 1`
+Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
+<!--ID: 1709044103781-->
+END%%
+
+### Shifting
 
 Left shift operations (`<<`) drop the `k` most significant bits and fills the right end of the result with `k` zeros. Right shift operations (`>>`) are classified in two ways:
 
@@ -1006,290 +1394,6 @@ Tags: c17
 <!--ID: 1707873410780-->
 END%%
 
-## Arithmetic
-
-### Addition
-
-Addition of two unsigned or two two's-complement numbers operate in much the same way as grade-school arithmetic. Digits are added one-by-one and overflows "carried" to the next summation. Overflows are truncated; the final carry bit is discarded in the underlying bit adder.
-
-%%ANKI
-Basic
-*Why* is adding $w$-bit integral types equal to $w$-bit truncation?
-Back: The underlying bit adder discards any final carry bit.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678721-->
-END%%
-
-%%ANKI
-Basic
-Why should you generally prefer `x < y` over `x - y < 0`?
-Back: The former avoids possible underflows.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678725-->
-END%%
-
-%%ANKI
-Basic
-How is `x - y < 0` rewritten more safely?
-Back: `x < y`
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678728-->
-END%%
-
-%%ANKI
-Basic
-What hardware-level advantage does two's-complement introduce over other signed encodings?
-Back: The same circuits can be used for unsigned and two's-complement addition.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678732-->
-END%%
-
-%%ANKI
-Basic
-What representational-level advantage does two's-complement introduce over other signed encodings?
-Back: `0` is encoded in only one way.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678736-->
-END%%
-
-Unsigned addition of $w$-bit integral types, denoted $+_w^u$, behaves like so:
-
-$$x +_w^u y = \begin{cases}
-x + y - 2^w & \text{if } x + y \geq 2^w \\
-x + y & \text{otherwise}
-\end{cases}$$
-
-This is more simply expressed as $x +_w^u y = (x + y) \bmod 2^w$.
-
-%%ANKI
-Basic
-What kind of overflows does unsigned addition potentially exhibit?
-Back: Positive overflow.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678739-->
-END%%
-
-%%ANKI
-Basic
-Why is unsigned addition overflow *not* UB?
-Back: Because the C standard enforces unsigned encoding of `unsigned` data types.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-Tags: c17
-<!--ID: 1708799678742-->
-END%%
-
-%%ANKI
-Basic
-What does $+_w^u$ denote?
-Back: Unsigned addition of $w$-bit integral types.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678745-->
-END%%
-
-%%ANKI
-Basic
-Unsigned addition overflow is equivalent to what bit-level manipulation tactic?
-Back: Truncation.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678748-->
-END%%
-
-%%ANKI
-Basic
-What is the result of $x +_w^u y$?
-Back: $(x + y) \bmod 2^w$
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678751-->
-END%%
-
-%%ANKI
-Basic
-*Why* does $x +_w^u y = (x + y) \bmod 2^w$?
-Back: Because discarding any carry bit is equivalent to truncating the sum to $w$ bits.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678755-->
-END%%
-
-%%ANKI
-Cloze
-Without using modular arithmetic, $x +_w^u y =$ {$x + y$} if {$x + y < 2^w$}.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678758-->
-END%%
-
-%%ANKI
-Cloze
-Without using modular arithmetic, $x +_w^u y =$ {$x + y - 2^w$} if {$x + y \geq 2^w$}.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678761-->
-END%%
-
-%%ANKI
-Basic
-How do you detect whether unsigned addition $s \coloneqq x +_w^u y$ overflowed?
-Back: Overflow occurs if and only if $s < x$.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678765-->
-END%%
-
-%%ANKI
-Basic
-How would you complete the body of this function?
-```c
-/* Determine whether arguments can be added without overflow */
-int uadd_ok(unsigned x, unsigned y);
-```
-Back:
-```c
-return (x + y) >= x;
-```
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678769-->
-END%%
-
-%%ANKI
-Basic
-Does unsigned overflow detection depend on the left or right operand of $s \coloneqq x +_w^u y$?
-Back: Either.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678772-->
-END%%
-
-%%ANKI
-Basic
-Why can we compare $s$ to $x$ or $y$ when detecting overflow of $s \coloneqq x +_w^u y$?
-Back: Because unsigned addition is commutative.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708799678776-->
-END%%
-
-%%ANKI
-Basic
-Given integer $0 < x < 2^w$, what is $x$'s unsigned additive inverse?
-Back: $2^w - x$
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708808252010-->
-END%%
-
-%%ANKI
-Basic
-Which unsigned integer is its own additive inverse?
-Back: $0$
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708808252017-->
-END%%
-
-Two's-complement addition, denoted $+_w^t$ operates similarly:
-
-$$x +_w^u y = \begin{cases}
-x + y - 2^w & \text{if } x + y \geq 2^{w-1} \\
-x + y + 2^w & \text{if } x + y < -2^{w-1} \\
-x + y & \text{otherwise}
-\end{cases}$$
-
-Unlike with unsigned addition, there is no simpler modulus operation that can be applied.
-
-%%ANKI
-Basic
-What kind of overflows does two's-complement addition potentially exhibit?
-Back: Positive and negative overflow.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376220-->
-END%%
-
-%%ANKI
-Basic
-Why is two's-complement addition overflow UB?
-Back: Because the C standard does not mandate any particular signed integer encoding.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-Tags: c17
-<!--ID: 1708964376225-->
-END%%
-
-%%ANKI
-Basic
-What does $+_w^t$ denote?
-Back: Two's-complement addition of $w$-bit integral types.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376228-->
-END%%
-
-%%ANKI
-Basic
-*Why* doesn't two's-complement addition perform modular arithmetic?
-Back: Because negative values are representable.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376231-->
-END%%
-
-%%ANKI
-Cloze
-$x +_w^t y =$ {$x + y - 2^w$} if {$x + y \geq 2^{w-1}$}.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376235-->
-END%%
-
-%%ANKI
-Cloze
-$x +_w^t y =$ {$x + y + 2^w$} if {$x + y < -2^{w-1}$}.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376238-->
-END%%
-
-%%ANKI
-Cloze
-$x +_w^t y =$ {$x + y$} if {$-2^{w-1} \leq x + y < 2^{w-1}$}.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376242-->
-END%%
-
-%%ANKI
-Basic
-How do we detect $x +_w^t y$ positive overflowed?
-Back: This happens iff $x > 0$, $y > 0$, and $x +_w^t y \leq 0$.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376246-->
-END%%
-
-%%ANKI
-Basic
-How do we detect $x +_w^t y$ negative overflowed?
-Back: This happens iff $x < 0$, $y < 0$, and $x +_w^t y \geq 0$.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376250-->
-END%%
-
-%%ANKI
-Basic
-How can we write $x +_w^t y$ in terms of unsigned addition?
-Back: $x +_w^t y = U2T_w(T2U_w(x) +_w^u T2U_w(y))$
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376254-->
-END%%
-
-%%ANKI
-*Why* are we able to characterize $+_w^t$ in terms of $+_w^u$?
-Back: Because two's-complement addition has the same bit-level representation as unsigned addition.
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-END%%
-
-%%ANKI
-Basic
-How would you complete the body of this function?
-```c
-/* Determine whether arguments can be added without overflow */
-int tadd_ok(int x, int y);
-```
-Back:
-```c
-int pos_over = x > 0 && y > 0 && (x + y) <= 0;
-int neg_over = x < 0 && y < 0 && (x + y) >= 0;
-return !pos_over && !neg_over;
-```
-Reference: Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.
-<!--ID: 1708964376259-->
-END%%
-
 ## References
 
 * Bryant, Randal E., and David O'Hallaron. *Computer Systems: A Programmer's Perspective*. Third edition, Global edition. Always Learning. Pearson, 2016.

notes/c17/strings.md

@@ -46,6 +46,15 @@ END%%
 
 The syntax for the format placeholder is `%[flags][width][.precision][length]specifier`.
 
+%%ANKI
+Basic
+What four optional parts make up a `printf` argument?
+Back: Flags, width, precision, and length.
+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: 1708974221761-->
+END%%
+
 %%ANKI
 Basic
 Which header file contains basic `printf` functionality?

notes/combinatorics/combinations.md

@@ -528,6 +528,38 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1708368078753-->
 END%%
 
+%%ANKI
+Basic
+What is a degree-0 polynomial?
+Back: A constant.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221746-->
+END%%
+
+%%ANKI
+Basic
+What name is given to a degree-0 polynomial?
+Back: A constant.
+Reference: Thomas H. Cormen et al., *Introduction to Algorithms*, 3rd ed (Cambridge, Mass: MIT Press, 2009).
+<!--ID: 1708974221749-->
+END%%
+
+%%ANKI
+Basic
+What name is given to a degree-1 polynomial?
+Back: A monomial.
+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: 1708974221752-->
+END%%
+
+%%ANKI
+Basic
+What name is given to a degree-2 polynomial?
+Back: A binomial.
+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: 1708974221755-->
+END%%
+
 %%ANKI
 Basic
 What is a binomial?

notes/combinatorics/inclusion-exclusion.md

@@ -45,7 +45,7 @@ END%%
 
 %%ANKI
 Basic
-What concept does PIE typically refer to?
+What concept does PIE refer to?
 Back: The **p**rinciple of **i**nclusion/**e**xclusion.
 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: 1708438356477-->

notes/combinatorics/permutations.md

@@ -64,7 +64,7 @@ END%%
 %%ANKI
 Basic
 How is $n!$ written recursively?
-Back: As $n(n - 1)!$.
+Back: $n(n - 1)!$
 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: 1708451749781-->
 END%%