A Very Interesting Problem
<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-a-very-interesting-problem" target="_blank">A Very Interesting Problem Exercise</a></p><p>There is a <strong><span style="color:#8e44ad;">Very Interesting Problem</span></strong> that illustrates how clever some of these "permutation with repeats" problems can get.</p>
<p><strong><span style="color:#27ae60;">The Problem</span></strong></p>
<p>Imagine there is a small car located at the origin of the $xy$-coordinate plane. The car can move in one-unit increments. How many distinct <strong><u>shortest</u></strong> routes can be taken by this car to the point $(5,4)$?</p>
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<p style="text-align: center;"><em>One possible route the car take</em></p>
<p><strong><span style="color:#e74c3c;">Check This Out</span></strong></p>
<p>Notice that we can map out the car's route using letters: U for an "up" move and R for a "right" one.</p>
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<p>And every single time, no matter the route, we have to move UP four times and move to the RIGHT five times. So we have this word: UUUURRRRR. </p>
<p>Now we know how to solve it! In how many ways can you distinctly arrange the letters UUUURRRRR?</p>
<p><strong><span style="color:#27ae60;">$$\frac{9!}{4!5!} = 126$$</span></strong></p>