<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-inclusion-exclusion-principle" target="_blank">Inclusion-Exclusion Principle Exercise</a></p><p>There's this awesome formula known as the <strong><span style="color:#8e44ad;">Inclusion-Exclusion Principle</span></strong>. Imagine we have two sets, Set $A$ and Set $B$. We can use the formula below to solve GRE-type problems.</p>
<p><span style="font-size:22px;">$$total = |A| + |B| - both + neither$$</span></p>
<p>Recall that those vertical lines in the case of sets don't mean absolute value. Rather, they mean the number of elements in Set $A$ or the number of elements in Set $B$. Honestly though, I usually just write it like this. Be a rebel.</p>
<p>$$total = A + B - both + neither$$</p>
<p><strong><span style="color:#e74c3c;">Example 1</span></strong></p>
<p>In a certain classroom of $46$ students, $40$ bring a backpack, $24$ bring a lunchbox, and $4$ bring neither. How many bring both?</p>
<p>$$46 = 40 + 24 - both + 4$$</p>
<p>$$46 = 68 - both$$</p>
<p>$$22 = both$$</p>
<p><strong><span style="color:#e74c3c;">Example 2</span></strong></p>
<p>A certain school with $120$ students offers only two foreign language classes: German and French. If $70$ of the students are studying German and $90$ are studying French, with $12$ studying neither, how many students at the school are learning only one foreign language?</p>
<p><u><strong>Note</strong></u>: Read the question above carefully. They're asking us to calculate the sum of the number of students studying ONLY German (not French too) and the number of students studying ONLY French (not German too).</p>
<p>$$120 = 70 + 90 - both + 12$$</p>
<p>$$both = 52$$</p>
<p>If $52$ are studying both, that means that $70 - 52 = 18$ students are studying only German and $90 - 52 = 38$ students are studying only French, so we get:</p>
<p>$$18 + 38 = 56$$</p>
<p style="text-align: center;"><em>Students studying only one foreign language (not both)</em></p>