Add fall midterms

src/content/questions/comp2804/midterm-fall-2014/1/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/1
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/1/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/1/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Let $n \geq 2$ be an integer. How many bitstrings of length $n$ are there that contain at least two
+		1s?
+`;
+
+const label1 = "${n \\choose 2} \\cdot 2^{n-2}$";
+const label2 = "$n \\cdot 2^{n-1}$";
+const label3 = "$2^{n} - 1 - n$";
+const label4 = "$2^{n} - n$";
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: true },
+    { label: label4, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/10/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/10
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/10/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/10/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+The function $f : \mathbb{N} \rightarrow \mathbb{N}$ is defined by
+		$$
+			\begin{align}
+			f(0) &= 8 \\
+			f(n) &= f(n - 1) + 6n - 8\; \ \mathrm{for}\ n \geq 1
+			\end{align}
+		$$
+		What is $f(n)$?
+`;
+
+const label1 = String.raw`$f(n) = 2n^{2} - 5n + 7$`;
+const label2 = String.raw`$f(n) = 2n^{2} - 5n + 8$`;
+const label3 = String.raw`$f(n) = 3n^{2} - 5n + 8$`;
+const label4 = String.raw`$f(n) = 4n^{2} - 5n + 8$`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: true },
+    { label: label4, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/11/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/11
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/11/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/11/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Let $B_n$ be the number of bitstrings of length $n$ that do not contain 1111. Which of the following
+		is true?
+`;
+
+const label1 = String.raw`$B_n = B_{n-1} + B_{n-2} + B_{n-3}$`;
+const label2 = String.raw`$B_n = B_{n-1} + B_{n-2} + B_{n-3} + B_{n-4}$`;
+const label3 = String.raw`$B_n = 2^{n} - 2^{n-4}$`;
+const label4 = String.raw`$B_n = 2^{n} - (n-3) \cdot 2^{n-4}$`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: true },
+    { label: label3, correct: false },
+    { label: label4, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/12/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/12
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/12/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/12/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+<div hidden>
+			$\newcommand{\Fib}{{\rm F \scriptsize IB}}$
+		</div>
+		
+		Consider the following recursive algorithm $\Fib$, which takes as input an
+		integer $n \geq 0$:
+		
+		<p>
+			$\mathbf{Algorithm}\ \Fib(n)\mathrm{:}$ <br>
+			$\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$ <br>
+			$\mathbf{then}\ f = n$ <br>
+			$\mathbf{else}\ f = \Fib(n - 1) + \Fib(n - 2)$ <br>
+			$\mathbf{endif};$ <br>
+			$\mathbf{return}\ f$ <br>
+		</p>
+		
+		When running $\Fib(9)$, how many calls are there to $\Fib(4)$?
+`;
+
+const label1 = String.raw`6`;
+const label2 = String.raw`7`;
+const label3 = String.raw`8`;
+const label4 = String.raw`9`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: true },
+    { label: label4, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/13/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/13
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/13/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/13/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+<div hidden>
+			$\newcommand{\Silly}{{\rm S \scriptsize ILLY}}
+			 \newcommand{\elsesp}{\phantom{\mathbf{else}\ }}$
+		</div>
+		
+		Consider the following recursive algorithm $\Silly$, which takes as input an integer $n \geq 1$ which is a power of 2:
+		
+		<p>
+			$\mathbf{Algorithm}\ \Silly(n)\mathrm{:}$ <br>
+			$\mathbf{if}\ n = 1$ <br>
+			$\mathbf{then}\ \text{drink one pint of beer}$ <br>
+			$\mathbf{else}\ \mathbf{if}\ n = 2$ <br>
+			$\elsesp \mathbf{then}\ \text{fart once}$ <br>
+			$\elsesp \mathbf{else}\ \text{fart once};$ <br>
+			$\elsesp \elsesp \Silly(n / 2);$ <br>
+			$\elsesp \elsesp \text{fart once}$ <br>
+			$\elsesp \mathbf{endif}$ <br>
+			$\mathbf{endif}$
+		</p>
+		
+		For $n$ a power of 2, let $F(n)$ be the number of times you fart when running algorithm $\Silly(n)$.
+		Which of the following is true? <br>
+		(n.b., $\log$ denotes the base-2 logarithm)
+`;
+
+const label1 = String.raw`$F(n) = \log n\ \mathrm{for}\ n \geq 1.$`;
+const label2 = String.raw`$F(n) = 2 \log n\ \mathrm{for}\ n \geq 1.$`;
+const label3 = String.raw`$F(n) = (2 \log n) - 1\ \mathrm{for}\ n \geq 1.$`;
+const label4 = String.raw`$F(n) = (2 \log n) - 1\ \mathrm{for}\ n \geq 2.$`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: false },
+    { label: label4, correct: true },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/14/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/14
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/14/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/14/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+You flip a fair coin 5 times. What is the probability that the first flip results in heads or the
+		fifth flip results in heads?
+`;
+
+const label1 = String.raw`1/4`;
+const label2 = String.raw`1/2`;
+const label3 = String.raw`3/4`;
+const label4 = String.raw`1`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: true },
+    { label: label4, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/15/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/15
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/15/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/15/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+You flip a fair coin 5 times. Define the events
+		[defs]
+		  A = "the number of heads is odd"
+		[/defs]
+		and
+		[defs]
+		  B = "the number of tails is even".
+		[/defs]
+		Which of the following is true?
+`;
+
+const label1 = String.raw`$\Pr(A) = \Pr(B)$`;
+const label2 = String.raw`$\Pr(A) < \Pr(B)$`;
+const label3 = String.raw`$\Pr(A) > \Pr(B)$`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: true },
+    { label: label2, correct: false },
+    { label: label3, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/16/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/16
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/16/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/16/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Consider 10 boxes and 10 balls. We throw each ball, uniformly, in a random box. What is the
+		probability that, after we have thrown all 10 balls, none of the 10 boxes is empty?
+`;
+
+const label1 = String.raw`$\frac{10!}{10^{10}}$`;
+const label2 = String.raw`$\frac{10^{10}}{10!}$`;
+const label3 = String.raw`$1 - \frac{10 \cdot (9/10)^{10}}{10^{10}}$`;
+const label4 = String.raw`None of the above.`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: true },
+    { label: label2, correct: false },
+    { label: label3, correct: false },
+    { label: label4, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/17/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/17
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/17/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/17/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Assume you write a multiple-choice exam that consists of 100 questions. For each question, 4 options
+		are given, one of which is the correct one. If you answer each of the 100 questions by choosing an
+		answer uniformly at random, what is the probability that you have exactly one correct answer?
+`;
+
+const label1 = String.raw`$\frac{100}{4^{100}}$`;
+const label2 = String.raw`$\frac{3^{99}}{4^{100}}$`;
+const label3 = String.raw`$\frac{100 + 3^{99}}{4^{100}}$`;
+const label4 = String.raw`$\frac{100 \cdot 3^{99}}{4^{100}}$`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: false },
+    { label: label4, correct: true },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/2/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/2
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/2/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/2/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+How many bitstrings of length 99 are there that start with 1010 or end with 1010?
+`;
+
+const label1 = String.raw`$2^{91}$`;
+const label2 = String.raw`$2^{95} + 2^{95}$`;
+const label3 = String.raw`$2^{96} - 2^{91}$`;
+const label4 = String.raw`$2^{99} - 2^{96}$`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: true },
+    { label: label4, correct: false },
+  ],
+};

src/content/questions/comp2804/midterm-fall-2014/3/index.md

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+---
+title: N/A
+path: comp2804/midterm-fall-2014/3
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-fall-2014/3/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-fall-2014/3/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Consider 17 boys and 17 girls. How many ways are there to arrange them on a line if all boys are
+		standing next to each other and all girls are standing next to each other?
+`;
+
+const label1 = String.raw`$17! + 17!$`;
+const label2 = String.raw`$2(17! + 17!)$`;
+const label3 = String.raw`$(17!)^2$`;
+const label4 = String.raw`$2(17!)^2$`;
+
+export const question: MultipleChoiceQuestion = {
+  body: body,
+  options: [
+    { label: label1, correct: false },
+    { label: label2, correct: false },
+    { label: label3, correct: false },
+    { label: label4, correct: true },
+  ],
+};