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<h2 class="hd hd-2 unit-title">Introduction</h2>
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<p>Read <a href="/assets/courseware/v1/b39f5889266425b78b7bc897e0a45ab5/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_Session27.pdf" target="[object Object]">Chapter 14.8 (PDF)</a> of<em> Mathematics for Computer Science</em> for 3.5 Pigeonhole Principle, Inclusion-Exclusion.</p>
<p>View the <a href="/assets/courseware/v1/ef56ace30d363d9b7799f032cd198143/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS15_cp27.pdf" target="[object Object]">Section 3.5 In-Class Questions (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Lecture Video | The Pigeonhole Principle</h2>
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<h2 class="hd hd-2 unit-title">Exercise | Rolling Dice</h2>
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<h2 class="hd hd-2 unit-title">Lecture Video | Inclusion-Exclusion Example</h2>
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<h2 class="hd hd-2 unit-title">Lecture Video | Inclusion-Exclusion 2 Sets</h2>
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<h3 class="hd hd-2">Inclusion-Exclusion 2 Sets</h3>
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<p>Download a copy of the slides for <a href="/assets/courseware/v1/8f7b8e324e3cbce369eab67c000bf7f2/asset-v1:OCW+6.042J+2T2019+type@asset+block/MIT6_042JS16_InclExclEx.pdf" target="[object Object]">Inclusion-Exclusion 2 Set Proof (PDF)</a></p>
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<h2 class="hd hd-2 unit-title">Exercise | Pigeonhole Principle</h2>
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Pigeonhole Principle
</h3>
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<p>Below is a list of properties that a group of people might possess.</p>
<p/>
<p>For each property, either give the minimum number of people that must be in a group to ensure that the property holds, or else write <strong>nh</strong> to indicate that the property need <strong>n</strong>ot <strong>h</strong>old even for arbitrarily large groups of people.</p>
<p/>
<p><em>Assume that every year has exactly 365 days; ignore leap years.</em></p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p> 1. At least 2 people were born on the same day of the year (ignore the year of birth). </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><p> 2. At least 2 people were born on January 1st. </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><p> 3. At least 3 people were born on the same day of the week.</p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><p> 4. At least 4 people were born in the same month. </p>
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 5" role="group"><p> 5. At least 2 people were born exactly one week apart. </p>
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<h2 class="hd hd-2 unit-title">Exercise | 6.042 TEAL Table</h2>
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6.042 TEAL Table
</h3>
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<div>
A 6.042 problem-solving team has 8 students (Alyssa, Ben, Carlos, ...) seated around a circular table. Since there are whiteboards all around, it doesn't matter which way students face, so two seatings are regarded as defining the same arrangement if each student has the same student on their right in both seatings. We'll be interested in counting how many arrangements there are of these 8 students, given some restrictions.
<br/>
<br/>
<em>Either input the final result or an equivalent form, using fact(x) to denote <ns0:math xmlns:ns0="http://www.w3.org/1998/Math/MathML">
<ns0:mi>x</ns0:mi>
<ns0:mo>!</ns0:mo>
</ns0:math>, explictly write out multiplication with *, and ** for exponentiation.</em>
<br/>
<br/>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="inputtype_8c7cc9f814e04b5696dd76d3efafe6d5_2_1" class=" capa_inputtype textline">
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<h2 class="hd hd-2 unit-title">Exercise | Class Schedules</h2>
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Class Schedules
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