Polynomials and delta-constants.

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

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notes/_journal/2024-04-18.md

@@ -1,11 +0,0 @@
----
-title: "2024-04-18"
----
-
-- [x] Anki Flashcards
-- [x] KoL
-- [ ] Sheet Music (10 min.)
-- [ ] Go (1 Life & Death Problem)
-- [ ] Korean (Read 1 Story)
-- [ ] Interview Prep (1 Practice Problem)
-- [ ] Log Work Hours (Max 3 hours)

notes/_journal/2024-04-19.md

@@ -0,0 +1,15 @@
+---
+title: "2024-04-19"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [x] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)
+
+* Hide-and-seek application
+	* Revisions to settings. Primarily flattening the structure.
+* Read 2.3 "Polynomial Fitting" of "Discrete Mathematics: An Open Introduction".

notes/_journal/2024-04/2024-04-18.md

@@ -0,0 +1,15 @@
+---
+title: "2024-04-18"
+---
+
+- [x] Anki Flashcards
+- [x] KoL
+- [ ] Sheet Music (10 min.)
+- [ ] Go (1 Life & Death Problem)
+- [ ] Korean (Read 1 Story)
+- [x] Interview Prep (1 Practice Problem)
+- [x] Log Work Hours (Max 3 hours)
+
+* Discussed first chapter of "Grokking the System Design Interview" as well as chapter 6 of "Designing Data-Intensive Applications" with Mike.
+* Applied to several more job openings.
+* Revised presence tracking and host updates in hide-and-seek application.

notes/algebra/polynomials.md

@@ -0,0 +1,130 @@
+---
+title: Polynomials
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algebra::polynomial
+tags:
+  - algebra
+  - polynomial
+---
+
+## Overview
+
+Given nonnegative integer $d$, a **polynomial in $n$ of degree $d$** is a function $p(n)$ of the form $$p(n) = \sum_{i=0}^d a_i n^i$$
+The coefficients of $p(n)$ are $a_0, a_1, \ldots, a_d$. Furthermore, $a_d \neq 0$.
+
+%%ANKI
+Basic
+Using sigma notation, a polynomial in $n$ of degree $d$ is a function of what form?
+Back: $p(n) = \sum_{i=0}^d a_in^i$ where $a_d \neq 0$.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713580808758-->
+END%%
+
+%%ANKI
+Basic
+What four algebraic operations are permitted in a polynomial?
+Back: Addition, subtraction, multiplication, and exponentiation.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713580808763-->
+END%%
+
+%%ANKI
+Basic
+What is $d$ in "a polynomial in $n$ of degree $d$"?
+Back: $d$ is a nonnegative integer.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713580808766-->
+END%%
+
+%%ANKI
+Basic
+What is $n$ in "a polynomial in $n$ of degree $d$"?
+Back: The polynomial's indeterminate.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713580808769-->
+END%%
+
+%%ANKI
+Basic
+Which coefficient is special in a polynomial and why?
+Back: That attached to the monomial with highest degree because it cannot be zero.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713580808772-->
+END%%
+
+%%ANKI
+Basic
+What are the coefficients of a polynomial?
+Back: The constants of the monomials found in the polynomial.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--ID: 1713580808776-->
+END%%
+
+%%ANKI
+Basic
+What name is given to a degree-0 polynomial?
+Back: A constant.
+Reference: Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).
+<!--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?
+Back: A polynomial containing two terms.
+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: 1708368078759-->
+END%%
+
+%%ANKI
+Basic
+What is polynomial fitting?
+Back: The solving of a linear system to find the coefficients of a polynomial.
+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: 1713580109018-->
+END%%
+
+%%ANKI
+Basic
+Polynomial fitting is a strategy for discovering what part of a polynomial?
+Back: The coefficients.
+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: 1713580808780-->
+END%%
+
+%%ANKI
+Basic
+What is the degree of a polynomial?
+Back: The highest degree of the monomials with non-zero coefficients.
+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: 1713580109082-->
+END%%
+
+%%ANKI
+Basic
+How many data points are required to fit a polynomial?
+Back: $k + 1$ where $k$ is the degree of the polynomial.
+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: 1713580109089-->
+END%%
+
+## Bibliography
+
+* 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).
+* Thomas H. Cormen et al., Introduction to Algorithms, Fourth edition (Cambridge, Massachusett: The MIT Press, 2022).

notes/algebra/sequences/arithmetic.md

@@ -150,6 +150,14 @@ Reference: Oscar Levin, *Discrete Mathematics: An Open Introduction*, 3rd ed., n
 <!--ID: 1709666305449-->
 END%%
 
+%%ANKI
+Basic
+Why is a sequence of partial sums named the way it is?
+Back: Each term is found by adding a finite number of infinite terms. 
+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: 1713580109297-->
+END%%
+
 ## Bibliography
 
 * 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).

notes/algebra/sequences/delta-constant.md

@@ -0,0 +1,201 @@
+---
+title: ∆ᵏ-constant Sequence
+TARGET DECK: Obsidian::STEM
+FILE TAGS: algebra::sequence algebra::polynomial
+tags:
+  - algebra
+  - sequence
+---
+
+## Overview
+
+Let $(a_n)$ be a sequence. We define the **$k$th differences of $(a_n)$** recursively:
+
+* The $0$th differences of $(a_n)$ is $(a_n)$.
+* The $k$th differences of $(a_n)$ is the sequence given by subtracting consecutive terms of the $(k-1)st$ differences of $(a_n)$.
+
+A sequence is said to be **$\Delta^k$-constant** if the $k$th differences are constant.
+
+> The closed formula for a sequence will be a degree $k$ polynomial if and only if the sequence is $\Delta^k$-constant.
+
+This is the discrete analogue to (continuous) derivatives of polynomials.
+
+%%ANKI
+Basic
+What are the $0$th differences of $(a_n)_{n \geq 0}$?
+Back: $(a_n)_{n \geq 0}$
+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: 1713580109096-->
+END%%
+
+%%ANKI
+Basic
+What are the $1$st differences of $(a_n)_{n \geq 0}$?
+Back: $(b_n)_{n \geq 1}$ where $b_n = a_n - a_{n - 1}$.
+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: 1713580109118-->
+END%%
+
+%%ANKI
+Basic
+How do we refer to the $1$st differences of $(a_n)$ more naturally?
+Back: As "the differences of $(a_n)$."
+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: 1713580109125-->
+END%%
+
+%%ANKI
+Basic
+What is the base case of the recursive definition of the $k$th differences of $(a_n)$?
+Back: $k = 0$
+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: 1713580109133-->
+END%%
+
+%%ANKI
+Basic
+What is the recurrence of the recursive definition of the $(k + 1)$st differences of $(a_n)$?
+Back: The $(k + 1)$st differences is the differences of the $k$th differences.
+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: 1713580109153-->
+END%%
+
+%%ANKI
+Basic
+What does it mean for $(a_n)$ to be $\Delta^k$-constant?
+Back: The $k$th differences of $(a_n)$ is constant.
+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: 1713580109175-->
+END%%
+
+%%ANKI
+Basic
+What can be said about the closed formula of a $\Delta^k$-constant sequence?
+Back: It is a polynomial with degree $k$.
+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: 1713580109182-->
+END%%
+
+%%ANKI
+Cloze
+We say a sequence is {$\Delta^k$-constant} when the {$k$th differences is constant}.
+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: 1713580109189-->
+END%%
+
+%%ANKI
+Basic
+$\Delta^k$-constant sequences are a discrete analogue to what calculus concept?
+Back: Derivatives.
+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).
+Tags: calculus
+<!--ID: 1713580109197-->
+END%%
+
+%%ANKI
+Basic
+What kind of mathematical expression do $\Delta^k$-constant sequences relate to?
+Back: Polynomials.
+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).
+Tags: calculus
+<!--ID: 1713580109203-->
+END%%
+
+%%ANKI
+Basic
+How can we prove every quadratic sequence, say $(a_n)$, has arithmetic differences?
+Back: By showing $a_{n+1} - a_n$ is linear.
+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: 1713580109225-->
+END%%
+
+%%ANKI
+Basic
+How can we prove every cubic sequence, say $(a_n)$, has quadratic differences?
+Back: By showing $a_{n+1} - a_n$ is quadratic.
+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: 1713580109232-->
+END%%
+
+%%ANKI
+Cloze
+The closed formula for a sequence will be a {degree $k$ polynomial} if and only if the $k$th differences {is constant}.
+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: 1713580109237-->
+END%%
+
+%%ANKI
+Basic
+How are arithmetic sequences defined in terms of $\Delta^k$ polynomials?
+Back: A sequence is arithmetic if and only if it is $\Delta^1$-constant.
+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: 1713580109244-->
+END%%
+
+%%ANKI
+Basic
+How are geometric sequences defined in terms of $\Delta^k$ polynomials?
+Back: N/A
+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: 1713580109250-->
+END%%
+
+%%ANKI
+Basic
+*Why* can't a geometric sequence be $\Delta^k$-constant for some $k \geq 0$?
+Back: Because the closed formula of a geometric sequence is not a polynomial.
+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: 1713580109255-->
+END%%
+
+%%ANKI
+Basic
+Suppose $(a_n)$ is a $\Delta^2$-constant sequence. What general form describes its closed formula?
+Back: $an^2 + bn + c$
+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: 1713580109261-->
+END%%
+
+%%ANKI
+Basic
+Suppose $(a_n)$ is a $\Delta^3$-constant sequence. What general form describes its closed formula?
+Back: $an^3 + bn^2 + cn + d$
+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: 1713580109265-->
+END%%
+
+%%ANKI
+Basic
+Suppose $(a_n)$ is a $\Delta^k$-constant sequence. Where is $k$ repeated in $(a_n)$'s closed formula?
+Back: At the largest degree of the polynomial.
+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: 1713580109270-->
+END%%
+
+%%ANKI
+Basic
+What are the terms in a $\Delta^0$-constant sequences?
+Back: $c, c, c, \ldots$ for some constant $c$.
+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: 1713580109274-->
+END%%
+
+%%ANKI
+Basic
+How do $\Delta^k$-constant sequences relate to polynomial fitting? 
+Back: The closed formula of a $\Delta^k$-constant sequence is a polynomial we can fit.
+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: 1713580109280-->
+END%%
+
+%%ANKI
+Basic
+How many data points of a $\Delta^k$-constant sequence are needed to polynomial fit its closed formula?
+Back: $k + 1$
+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: 1713580109285-->
+END%%
+
+## Bibliography
+
+* 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).

notes/algebra/sequences/triangular-numbers.md

@@ -254,6 +254,14 @@ Reference: “Triangular Number,” in _Wikipedia_, January 13, 2024, [https://e
 <!--ID: 1709419325981-->
 END%%
 
+%%ANKI
+Basic
+The triangular numbers is the sequence of partial sums of what sequence?
+Back: $a_n = n$
+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: 1713580109292-->
+END%%
+
 ## Bibliography
 
 * 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).

notes/combinatorics/combinations.md

@@ -525,46 +525,6 @@ 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).
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-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?
-Back: A polynomial containing two terms.
-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: 1708368078759-->
-END%%
-
 %%ANKI
 Basic
 Why are binomial coefficients named the way they are?