The Combinations Pattern

<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-the-combinations-pattern" target="_blank">The Combinations Pattern Exercise</a></p><p>Imagine we have seven items. In how many ways can we take zero of those items? One? Two? Three? Four? Five? Six? Seven?</p> <p>If you do all eight of these calculations with the combinations formula $\frac{n!}{r!(n-r)!}$, you&#39;ll get these eight numbers:</p> <p>$$1, 7, 21, 35, 35, 21, 7, 1$$</p> <p>Do you see the pattern? Do you see <strong><span style="color:#8e44ad;">The Combinations Pattern</span></strong>? The data is symmetrical. And this makes sense because choosing $2$ items from the seven results in the same formula as choosing $5$ items from the seven:</p> <p>$$\frac{7!}{2!5!} = \frac{7!}{5!2!}$$</p> <p><strong><span style="color:#27ae60;">The Difference Between Even and Odd&nbsp;</span></strong></p> <p>If the total number of elements is odd, as in our example above, the data results in two equal peaks in the middle. 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However, in the case of an even number of elements, say $6$, you get only one peak in the symmetry.&nbsp;</p>