Add midterm-winter-2019

src/content/evaluations/comp2804/midterm-winter-2019.md

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+---
+title: Midterm Winter 2019
+path: comp2804/midterm-winter-2019
+author: Michiel Smid
+course: comp2804
+questions:
+  - comp2804/midterm-winter-2019/1
+  - comp2804/midterm-winter-2019/2
+  - comp2804/midterm-winter-2019/3
+  - comp2804/midterm-winter-2019/4
+  - comp2804/midterm-winter-2019/5
+  - comp2804/midterm-winter-2019/6
+  - comp2804/midterm-winter-2019/7
+  - comp2804/midterm-winter-2019/8
+  - comp2804/midterm-winter-2019/9
+  - comp2804/midterm-winter-2019/10
+  - comp2804/midterm-winter-2019/11
+  - comp2804/midterm-winter-2019/12
+  - comp2804/midterm-winter-2019/13
+  - comp2804/midterm-winter-2019/14
+  - comp2804/midterm-winter-2019/15
+  - comp2804/midterm-winter-2019/16
+  - comp2804/midterm-winter-2019/17
+---

src/content/questions/comp2804/midterm-winter-2019/1/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/1
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/1/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/1/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Consider bitstrings of length 13. The positions in these strings are numbered as $1,2,3,\dots,13$.
+		How many such bitstrings have the property that all bits at the odd positions are equal?
+`;
+
+const label1 = String.raw`32`;
+const label2 = String.raw`64`;
+const label3 = String.raw`128`;
+const label4 = String.raw`256`;
+
+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-winter-2019/10/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/10
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/10/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/10/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+A bitstring is called 00-<em>free</em>, if it does not contain two 0's next to each other.
+		In class, we have seen that for any $m \geq 1$,
+		the number of 00-free bitstrings of length $m$ is equal to the $(m+2)$-th Fibonacci number
+		$f_{m+2}$. <br>
+		What is the number of 00-free bitstrings of length 77 that have 0 at position 59? (The positions are numbered $1,2,\dots,77$.)
+`;
+
+const label1 = String.raw`$f_{17} \cdot f_{57}$`;
+const label2 = String.raw`$f_{18} \cdot f_{58}$`;
+const label3 = String.raw`$f_{19} \cdot f_{59}$`;
+const label4 = String.raw`$f_{20} \cdot f_{60}$`;
+
+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-winter-2019/11/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/11
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/11/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/11/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 recursively defined as follows:
+		$$
+			\begin{align}
+			f(0) &= 6, \\
+			f(n) &= 4 \cdot f(n-1) + 2^{n} \ \ \mathrm{if}\ n \geq 1.
+			\end{align}
+		$$
+		Which of the following is true for all integers $n \geq 0$?
+`;
+
+const label1 = String.raw`$f(n) = 6 \cdot 4^{n} - 2^{n}$`;
+const label2 = String.raw`$f(n) = 7 \cdot 4^{n} - 2^{n}$`;
+const label3 = String.raw`$f(n) = 8 \cdot 4^{n} - 2^{n+1}$`;
+const label4 = String.raw`None of the above.`;
+
+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-winter-2019/12/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/12
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/12/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/12/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Consider strings of characters, where each character is an element of the set $\{a, b, c\}$. Such a
+		string is called <em>awesome</em>, if it does not contain $aa$, and does not contain $aba$, and does not contain $abb$. <br>
+		For any integer $n \geq 1$, let $A_n$ be the number of awesome strings of length $n$. <br>
+		Which of the following is true for any integer $n \geq 4$?
+`;
+
+const label1 = String.raw`$A_n = A_{n-1} + 2 \cdot A_{n-2} + 2 \cdot A_{n-3}$`;
+const label2 = String.raw`$A_n = A_{n-1} + 2 \cdot A_{n-2} + A_{n-3}$`;
+const label3 = String.raw`$A_n = 2 \cdot A_{n-1} + A_{n-2} + 2 \cdot A_{n-3}$`;
+const label4 = String.raw`$A_n = 2 \cdot A_{n-1} + A_{n-2} + A_{n-3}$`;
+
+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-winter-2019/13/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/13
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/13/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/13/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+We consider binary $2 \times n$ matrices, i.e., matrices with 2 rows and $n$ columns, in which each
+		entry is 0 or 1. A column in such a matrix is called a <span class="sticky">$1 \atop 1$-<em>column</em>,</span>
+		if both bits in the column are 1. <br>
+		For any integer $n \geq 1$ and integer $k$ with $0 \leq k \leq n$, let $M(n,k)$ be the number of binary $2 \times n$ matrices
+		<ul style="margin: 0.5rem 0;">
+			<li>
+				that do not contain any $1 \atop 1$-column, and
+			</li>
+			<li>
+				contain exactly $k$ many 1's.
+			</li>
+		</ul>
+		Which of the following is true for all <span class="sticky">integers $n \geq 2$</span> and $k$ with $1 \leq k \leq n-1$?
+`;
+
+const label1 = String.raw`$M(n,k) = M(n-1,k)\ +$$\ M(n-1,k-1)$`;
+const label2 = String.raw`$M(n,k) = M(n-1,k)\ +$$\ 2 \cdot M(n-1,k-1)$`;
+const label3 = String.raw`$M(n,k) = M(n,k-1)\ +$$\ M(n-1,k-1)$`;
+const label4 = String.raw`$M(n,k) = M(n,k-1)\ +$$\ 2 \cdot M(n-1,k-1)$`;
+
+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-winter-2019/14/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/14
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/14/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/14/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+<div hidden>$\newcommand{\HelloWorld}{{\rm H {\scriptsize ELLO} W {\scriptsize ORLD}}}$</div>
+		
+		Consider the recursive algorithm $\HelloWorld$, which takes as input an integer $n \geq 0$:
+		<p>
+			$\mathbf{Algorithm}\ \HelloWorld(n)\mathrm{:}$ <br>
+			$\mathbf{if}\ n = 0\ \mathrm{or}\ n = 1$ <br>
+			$\mathbf{then}\ \mathrm{print}\ \mathit{Hello}\ \mathit{World}$ <br>
+			$\mathbf{else}\ \mathbf{if}\ n \text{ is a multiple of } 3$ <br>
+			$\qquad \mathbf{then}\ \HelloWorld\left( n \middle/ 3 \right);$ <br>
+			$\qquad \qquad \ \ \mathrm{print}\ \mathit{Hello}\ \mathit{World};$ <br>
+			$\qquad \qquad \ \ \HelloWorld\left( 2n \middle/ 3 \right)$ <br>
+			$\qquad \mathbf{else}\ \HelloWorld(n + 1)$ <br>
+			$\qquad \mathbf{endif};$ <br>
+			$\mathbf{endif}$
+		</p>
+		Which of the following is correct?
+`;
+
+const label1 = String.raw`For any integer $n \geq 0$, algorithm $\HelloWorld(n)$ terminates.`;
+const label2 = String.raw`There exists an integer $n \geq 0$, for which algorithm $\HelloWorld(n)$ does not terminate.`;
+const label3 = String.raw`All of the above.`;
+const label4 = String.raw`None of the above.`;
+
+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-winter-2019/15/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/15
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/15/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/15/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+The Carleton Computer Science Society (CCSS) is organizing their annual Saint Patrick's Day party.
+		The CCSS has bought three types of drinks:
+		<ul style="margin: 0.5rem 0;">
+			<li>
+				Porterhouse Brewing Co. Oyster Stout,
+			</li>
+			<li>
+				Guinness Extra Stout,
+			</li>
+			<li>
+				Magners Original Irish Cider.
+			</li>
+		</ul>
+		There is an unlimited supply for each of these types. <br>
+		There are 75 students at the party, and each of them gets one drink, which is chosen uniformly at random from these three types. <br>
+		Let $A$ be the event
+		[defs]
+		  A = "exactly 50 students get Magners Original Irish Cider".
+		[/defs]
+		What is $\Pr(A)$?
+`;
+
+const label1 = String.raw`$\frac{{ 75 \choose 50 } \cdot 2^{25}}{3^{75}}$`;
+const label2 = String.raw`$\frac{3^{75}}{{75 \choose 50} \cdot 2^{25}}$`;
+const label3 = String.raw`$\frac{75 \choose 50}{3^{75}}$`;
+const label4 = String.raw`$\frac{{75 \choose 50} \cdot 3^{25}}{3^{75}}$`;
+
+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-winter-2019/16/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/16
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/16/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/16/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+Consider a standard 6-sided fair die. We roll this die 8 times. Let $A$ be the event
+		[defs]
+		  A = "there are at least two 5's in the sequence of 8 rolls".
+		[/defs]
+		What is $\Pr(A)$?
+`;
+
+const label1 = String.raw`$1 - {\frac{8 \cdot 5^{7}}{6^{8}}}$`;
+const label2 = String.raw`$1 - {\frac{6^{8}}{5^{8} + 8 \cdot 5^{7}}}$`;
+const label3 = String.raw`$1 - {\frac{5^{8} + 8 \cdot 5^{7}}{6^{8}}}$`;
+const label4 = String.raw`$1 - {\frac{8^{5} + 8 \cdot 7^{5}}{8^{6}}}$`;
+
+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-winter-2019/17/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/17
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/17/question.ts
+tags:
+  - comp2804
+---

src/content/questions/comp2804/midterm-winter-2019/17/question.ts

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+import type { MultipleChoiceQuestion } from "@common/MultipleChoiceQuestionGenerator";
+
+const body = String.raw`
+This midterm has 17 questions. For each question, four options are given, exactly one of which is
+		correct. Assume that you answer each question, by choosing one of the four options uniformly at random. <br>
+		Let A be the event
+		[defs]
+		  A = "you answer at least 16 questions correctly".
+		[/defs]
+		What is $\Pr(A)$?
+`;
+
+const label1 = String.raw`$\frac{4^{17}}{52}$`;
+const label2 = String.raw`$\frac{51}{4^{17}}$`;
+const label3 = String.raw`$\frac{49}{4^{17}}$`;
+const label4 = String.raw`$\frac{52}{4^{17}}$`;
+
+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-winter-2019/2/index.md

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+---
+title: N/A
+path: comp2804/midterm-winter-2019/2
+type: multiple-choice
+author: Michiel Smid
+question: comp2804/midterm-winter-2019/2/question.ts
+tags:
+  - comp2804
+---