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618 lines
25 KiB
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Factorial !</h1>
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<p class="center"><i>Example: <b>4!</b> is shorthand for <b>4 × 3 × 2 × 1</b></i></p>
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<div class="simple">
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td style="width:60px; text-align:center;"><img src="images/factorial.svg" alt="Factorial Symbol" height="" width=""></td>
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<td>
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<p>The <b>factorial function</b> (symbol: <b><font size="+1">!</font></b>) says to <b>multiply all whole numbers</b> from our chosen number down to 1.</p>
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<p>Examples:</p>
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<ul>
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<div class="bigul">
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<li><b>4!</b> = 4 × 3 × 2 × 1 = 24</li>
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<li><b>7!</b> = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5040</li>
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<li><b>1!</b> = 1</li>
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</div>
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</ul> </td>
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</tr>
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</tbody></table></div><br>
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<div class="words">We usually say (for example) <b> 4!</b> as "4 factorial", but some people say "4 shriek" or "4 bang"</div>
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<h2>Calculating From the Previous Value</h2>
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<p>We can easily calculate a factorial from the previous one:</p>
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<p class="center"><img src="images/factorial-how.svg" alt="factorial multiply" height="61" width="279"></p>
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<p>As a table:</p>
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<div class="beach">
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<table align="center" width="400" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th>n</th>
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<th>n!</th>
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<th> </th>
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<th> </th>
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</tr>
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<tr style="text-align:center;">
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<td>1</td>
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<td><b>1</b></td>
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<td>1</td>
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<td>1</td>
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</tr>
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<tr style="text-align:center;">
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<td>2</td>
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<td>2 × <b>1</b></td>
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<td>= 2 × <b>1!</b></td>
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<td>= 2</td>
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</tr>
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<tr style="text-align:center;">
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<td>3</td>
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<td>3 × <b>2 × 1</b></td>
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<td>= 3 × <b>2!</b></td>
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<td>= 6</td>
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</tr>
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<tr style="text-align:center;">
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<td>4</td>
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<td>4 × <b>3 × 2 × 1</b></td>
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<td>= 4 × <b>3!</b></td>
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<td>= 24</td>
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</tr>
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<tr style="text-align:center;">
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<td>5</td>
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<td>5 × <b>4 × 3 × 2 × 1</b></td>
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<td>= 5 × <b>4!</b></td>
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<td>= 120</td>
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</tr>
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<tr style="text-align:center;">
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<td>6</td>
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<td>etc</td>
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<td>etc</td>
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<td> </td>
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</tr>
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</tbody></table>
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</div><br>
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<ul>
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<li>To work out 6!, multiply 120 by <b>6</b> to get 720</li>
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<li>To work out 7!, multiply 720 by <b>7</b> to get 5040</li>
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<li>And so on</li>
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</ul>
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<div class="example">
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<h3>Example: 9! equals 362,880. Try to calculate 10!</h3>
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<p>10! = 10 × 9!</p>
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<p>10! = 10 × 362,880 = <b>3,628,800</b></p>
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</div>
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<p>So the rule is:</p>
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<p class="center large">n! = n × (n−1)!</p>
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<p>Which says</p>
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<p class="center">"the factorial of any number is <b>that number</b> times the <b>factorial of (that number minus 1)</b>"</p>
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<p>So 10! = 10 × 9!, ... and 125! = 125 × 124!, etc.</p>
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<h2>What About "0!"</h2>
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<p>Zero Factorial is interesting ... it is generally agreed that <b>0! = 1</b>.</p>
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<p>It may seem funny that multiplying no numbers together results in 1, but let's follow the pattern backwards from, say, 4! like this:</p>
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<p class="center"><img src="images/zero-factorial.svg" alt="24/4=6, 6/3=2, 2/2=1, 1/1=1" height="149" width="130"></p>
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<p>And in many equations using 0! = 1 just makes sense.</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/arrange-letters-acb.jpg" alt="arrange letters acb" height="113" width="200"></p>
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<h3>Example: how many ways can we arrange letters (without repeating)?</h3>
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<ul>
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<li>For 1 letter "a" there is only <b>1</b> way: a</li>
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<li>For 2 letters "ab" there are <b>1×2=2</b> ways: ab, ba</li>
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<li>For 3 letters "abc" there are <b>1×2×3=6</b> ways: abc acb cab bac bca cba</li>
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<li>For 4 letters "abcd" there are <b>1×2×3×4=24</b> ways: (try it yourself!)</li>
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<li>etc</li>
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</ul>
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<p>The formula is simply <b>n!</b></p>
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<p>Now ... how many ways can we arrange no letters? Just one way, an empty space:</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/arrange-letters-none.jpg" alt="arrange letters none" height="113" width="200"></p>
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<p>So <b>0! = 1</b></p>
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</div>
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<h2>Where is Factorial Used?</h2>
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<p>One area they are used is in <a href="../combinatorics/combinations-permutations.html">Combinations and Permutations</a>. We had an example above, and here is a slightly different example:</p>
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<div class="example">
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<p style="float:right; margin: 0 0 5px 10px;"><img src="images/winners.jpg" alt="1st, 2nd and 3rd" height="202" width="219"></p>
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<h3>Example: How many different ways can 7 people come 1<sup>st</sup>, 2<sup>nd</sup> and 3<sup>rd</sup>?</h3>
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<p>The list is quite long, if the 7 people are called <i>a,b,c,d,e,f</i> and <i>g</i> then the list includes:</p>
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<p class="center"><i>abc, abd, abe, abf, abg, acb, acd, ace, acf, ...</i> etc.</p>
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<p>The formula is <span class="intbl"><em>7!</em><strong>(7−3)!</strong></span><b> = <span class="intbl"><em>7!</em><strong>4!</strong></span></b></p>
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<p>Let us write the multiplies out in full:</p>
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<p class="center larger"><span class="intbl"><em style="text-align:right;"><b>7 × 6 × 5 × 4 × 3 × 2 × 1</b></em><strong style="text-align:right;"><b>4 × 3 × 2 × 1</b></strong></span><b> = 7 × 6 × 5</b></p>
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<p>That was neat. The <b>4 × 3 × 2 × 1</b> "cancelled out", leaving only <b>7 × 6 × 5</b>. And:</p>
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<p class="center larger"><b>7 × 6 × 5 = 210</b></p>
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<p>So there are 210 different ways that 7 people could come 1<sup>st</sup>, 2<sup>nd</sup> and 3<sup>rd</sup>.</p>
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<p>Done!</p>
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</div>
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<div class="example">
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<h3>Example: What is 100! / 98!</h3>
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<p>Using our knowledge from the previous example we can jump straight to this:</p>
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<p class="center larger"><span class="intbl"><em>100!</em><strong>98!</strong></span> = 100 × 99 = 9900</p>
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</div>
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<h2>A Small List</h2>
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<div class="beach">
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<table align="center" width="50%" border="0">
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<tbody>
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<tr style="text-align:center;">
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<th>n</th>
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<th>n!</th>
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</tr>
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<tr style="text-align:center;">
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<td>0</td>
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<td>1</td>
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</tr>
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<tr style="text-align:center;">
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<td>1</td>
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<td>1</td>
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</tr>
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<tr style="text-align:center;">
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<td>2</td>
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<td>2</td>
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</tr>
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<tr style="text-align:center;">
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<td>3</td>
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<td>6</td>
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</tr>
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<tr style="text-align:center;">
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<td>4</td>
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<td>24</td>
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</tr>
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<tr style="text-align:center;">
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<td>5</td>
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<td>120</td>
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</tr>
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<tr style="text-align:center;">
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<td>6</td>
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<td>720</td>
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</tr>
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<tr style="text-align:center;">
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<td>7</td>
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<td>5,040</td>
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</tr>
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<tr style="text-align:center;">
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<td>8</td>
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<td>40,320</td>
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</tr>
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<tr style="text-align:center;">
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<td>9</td>
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<td>362,880</td>
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</tr>
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<tr style="text-align:center;">
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<td>10</td>
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<td>3,628,800</td>
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</tr>
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<tr style="text-align:center;">
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<td>11</td>
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<td>39,916,800</td>
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</tr>
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<tr style="text-align:center;">
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<td>12</td>
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<td>479,001,600</td>
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</tr>
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<tr style="text-align:center;">
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<td>13</td>
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<td>6,227,020,800</td>
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</tr>
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<tr style="text-align:center;">
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<td>14</td>
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<td>87,178,291,200</td>
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</tr>
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<tr style="text-align:center;">
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<td>15</td>
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<td>1,307,674,368,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>16</td>
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<td>20,922,789,888,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>17</td>
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<td>355,687,428,096,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>18</td>
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<td>6,402,373,705,728,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>19</td>
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<td>121,645,100,408,832,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>20</td>
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<td>2,432,902,008,176,640,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>21</td>
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<td> 51,090,942,171,709,440,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>22</td>
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<td> 1,124,000,727,777,607,680,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>23</td>
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<td> 25,852,016,738,884,976,640,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>24</td>
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<td> 620,448,401,733,239,439,360,000</td>
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</tr>
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<tr style="text-align:center;">
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<td>25</td>
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<td> 15,511,210,043,330,985,984,000,000</td>
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</tr>
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</tbody></table></div>
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<p>As you can see, it gets big quickly.</p>
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<p>If you need more, try the <a href="../calculator-precision.html">Full Precision Calculator</a>.</p>
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<h2>Interesting Facts</h2>
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<div class="fun">
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<p><b>Six weeks</b> is exactly <b>10!</b> seconds (=3,628,800)</p>
|
||
<p>Here is why:</p>
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:right;"><i>Seconds in 6 weeks:</i></td>
|
||
<td style="width:20px;"> </td>
|
||
<td style="text-align:center;">60 × 60 × 24 × 7 × 6</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"><i>Factor some numbers:</i></td>
|
||
<td> </td>
|
||
<td style="text-align:center;">(2 × 3 × 10) × (3 × 4 × 5) × (8 × 3) × 7 × 6</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"><i>Rearrange:</i></td>
|
||
<td> </td>
|
||
<td style="text-align:center;">2 × 3 × 4 × 5 × 6 × 7 × 8 × 3 × 3 × 10</td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:right;"><i>Lastly 3×3=9:</i></td>
|
||
<td> </td>
|
||
<td style="text-align:center;">2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10</td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div><p> </p>
|
||
<div class="fun">
|
||
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/deck-cards.jpg" alt="deck of cards" height="148" width="200"></p>
|
||
<p>There are <b>52!</b> ways to shuffle a deck of cards.</p>
|
||
<p>That is <b>8.0658175... × 10<sup>67</sup></b></p>
|
||
<p>Just shuffle a deck of cards and it is likely that you are the <b>first person ever</b> with that particular order.</p>
|
||
</div><p> </p>
|
||
<div class="fun">
|
||
<p>There are about <b>60!</b> atoms in the observable Universe.</p>
|
||
<p>60! is about <b> 8.320987... × 10<sup>81</sup></b> and the current estimates are between 10<sup>78</sup> to 10<sup>82</sup> atoms in the observable Universe.</p>
|
||
</div>
|
||
<p> </p>
|
||
<div class="fun">
|
||
<p><b>70!</b> is approximately <b>1.197857... x 10<sup>100</sup></b>, which is just larger than a Googol (the digit 1 followed by one hundred zeros).</p>
|
||
<p><b>100!</b> is approximately 9.3326215443944152681699238856 x 10<sup>157</sup></p>
|
||
<p><b>200!</b> is approximately 7.8865786736479050355236321393 x 10<sup>374</sup></p>
|
||
</div>
|
||
<p> </p>
|
||
|
||
<h1>Advanced Topics</h1>
|
||
|
||
<h3>A Close Formula!</h3>
|
||
|
||
<!--
|
||
<div class="center large">n! ≈ n<sup>n+1</sup> e<sup>−n</sup>
|
||
<div class="intgl">
|
||
<div class="to">∞</div>
|
||
<div class="symb"></div>
|
||
<div class="from">0</div>
|
||
</div> e<sup>a<span class="intbl"><em>−n</em><strong>2</strong></span> (z−1)<sup>2</sup></sup> dz</div>
|
||
<!-- n! APR n^n+1 e^-n INT{0,INF} e^a -n/2 (z-1)^2 dz -->
|
||
|
||
|
||
<div class="center large">n! ≈ (<span class="intbl"><em>n</em><strong>e</strong></span>)<sup>n</sup> <span style="font-size:120%;">√</span><span class="overline">2<span class="times">π</span>n </span></div>
|
||
<!-- n! APR ( n/e )^n SQR( 2PIn ) -->
|
||
|
||
<p>The "≈" means "approximately equal to". Let us see how good it is: </p>
|
||
<div class="beach">
|
||
|
||
|
||
<table style="" class="center">
|
||
<tbody>
|
||
<tr><th>n</th><th>n!</th><th>Close Formula<br>(to 2 Decimals)</th><th>Accuracy<br>(to 4 Decimals)</th></tr>
|
||
|
||
<tr>
|
||
<td>1</td>
|
||
<td>1</td>
|
||
<td>0.92</td>
|
||
<td>0.9221</td></tr>
|
||
<tr>
|
||
<td>2</td>
|
||
<td>2</td>
|
||
<td>1.92</td>
|
||
<td>0.9595</td></tr>
|
||
<tr>
|
||
<td>3</td>
|
||
<td>6</td>
|
||
<td>5.84</td>
|
||
<td>0.9727</td></tr>
|
||
<tr>
|
||
<td>4</td>
|
||
<td>24</td>
|
||
<td>23.51</td>
|
||
<td>0.9794</td></tr>
|
||
<tr>
|
||
<td>5</td>
|
||
<td>120</td>
|
||
<td>118.02</td>
|
||
<td>0.9835</td></tr>
|
||
<tr>
|
||
<td>6</td>
|
||
<td>720</td>
|
||
<td>710.08</td>
|
||
<td>0.9862</td></tr>
|
||
<tr>
|
||
<td>7</td>
|
||
<td>5040</td>
|
||
<td>4980.40</td>
|
||
<td>0.9882</td></tr>
|
||
<tr>
|
||
<td>8</td>
|
||
<td>40320</td>
|
||
<td>39902.40</td>
|
||
<td>0.9896</td></tr>
|
||
<tr>
|
||
<td>9</td>
|
||
<td>362880</td>
|
||
<td>359536.87</td>
|
||
<td>0.9908</td></tr>
|
||
<tr>
|
||
<td>10</td>
|
||
<td>3628800</td>
|
||
<td>3598695.62</td>
|
||
<td>0.9917</td></tr>
|
||
<tr>
|
||
<td>11</td>
|
||
<td>39916800</td>
|
||
<td>39615625.05</td>
|
||
<td>0.9925</td></tr>
|
||
<tr>
|
||
<td>12</td>
|
||
<td>479001600</td>
|
||
<td>475687486.47</td>
|
||
<td>0.9931</td></tr>
|
||
|
||
|
||
|
||
</tbody></table>
|
||
</div>
|
||
<p>If you don't need perfect accuracy this may be useful.</p>
|
||
<p>Note: it is called "Stirling's approximation" and is based on a simplifed version of the <a href="gamma-function.html">Gamma Function</a>.</p>
|
||
<p><br></p>
|
||
<h3>What About Negatives?</h3>
|
||
<p>Can we have factorials for negative numbers?</p>
|
||
<p><b>Yes ... but not for n</b><b>egative <i>integers.</i></b></p>
|
||
<p>Negative <i>integer</i> factorials (like -1!, -2!, etc) are <b>undefined</b>.</p>
|
||
|
||
<p>Let's start with 3! = 3 × 2 × 1 = 6 and go <b>down</b>:</p>
|
||
|
||
<table style="border: 0;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="width:20px;"> </td>
|
||
<td style="text-align:right;">2!</td>
|
||
<td style="text-align:center; width:20px;">=</td>
|
||
<td style="text-align:center;">3! / 3</td>
|
||
<td style="text-align:center; width:20px;">=</td>
|
||
<td style="text-align:center;">6 / 3</td>
|
||
<td style="text-align:center; width:20px;">=</td>
|
||
<td>2</td>
|
||
<td style="width:30px;"> </td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;">1! </td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">2! / 2</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">2 / 2</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td>1</td>
|
||
<td><br></td>
|
||
<td> </td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;">0!</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">1! / 1</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">1 / 1</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td>1</td>
|
||
<td><br></td>
|
||
<td><i> which is why 0!=1</i></td>
|
||
</tr>
|
||
<tr>
|
||
<td> </td>
|
||
<td style="text-align:right;"> (−1)!</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">0! / 0 </td>
|
||
<td style="text-align:center;">=</td>
|
||
<td>1 / 0</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td>? </td>
|
||
<td><br></td>
|
||
<td><i><b> oops, dividing by zero is undefined</b></i></td>
|
||
</tr>
|
||
</tbody></table>
|
||
<p>And from here on down <b>all integer factorials are undefined</b>.</p>
|
||
<p><br></p>
|
||
|
||
<h3>What About Decimals?</h3>
|
||
<p>Can we have factorials for numbers like 0.5 or −3.217?</p>
|
||
|
||
<p><b>Yes we can!</b> But we need to use the <a href="gamma-function.html">Gamma Function</a> (advanced topic).</p>
|
||
<p>Factorials can also be negative (except for negative integers).</p>
|
||
<p><br></p>
|
||
|
||
<h3>Half Factorial</h3>
|
||
<p>But I can tell you the factorial of <b>half</b> (½) is <b>half of the square root of <a href="pi.html">pi</a></b> .</p>
|
||
<p>Here are some "half-integer" factorials:</p>
|
||
<div class="simple">
|
||
|
||
<table style="border: 0; margin:auto;">
|
||
<tbody>
|
||
<tr>
|
||
<td style="text-align:center; width:100px;">(−½)!</td>
|
||
<td style="text-align:center; width:30px;">=</td>
|
||
<td style="text-align:center; width:100px;"><span class="times">√π</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">(½)!</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">(½)<span class="times">√π</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">(3/2)!</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">(3/4)<span class="times">√π</span></td>
|
||
</tr>
|
||
<tr>
|
||
<td style="text-align:center;">(5/2)!</td>
|
||
<td style="text-align:center;">=</td>
|
||
<td style="text-align:center;">(15/8)<span class="times">√π</span></td>
|
||
</tr>
|
||
</tbody></table>
|
||
</div>
|
||
<p>It still follows the rule that "the factorial of any number is <b>that number times the factorial of (1 smaller than that number)</b>", because</p>
|
||
<p class="center">(3/2)! = (3/2) × (1/2)!<br>
|
||
(5/2)! = (5/2) × (3/2)!</p>
|
||
<p>Can you figure out what (7/2)! is?</p>
|
||
<p><br></p>
|
||
<h3>Double Factorial!!</h3>
|
||
<p>A double factorial is like a normal factorial but we skip every second number:</p>
|
||
<ul>
|
||
<li> 8!! = 8 × 6 × 4 × 2 = 384 </li>
|
||
<li>9!! = 9 × 7 × 5 × 3 × 1 = 945 </li></ul>
|
||
<p>Notice how we multiply all even, or all odd, numbers. </p>
|
||
<p>Note: if we want to apply factorial twice we write (n!)!</p>
|
||
|
||
<p> </p>
|
||
<div class="questions">2229, 2230, 7006, 2231, 7007, 9080, 9081, 9082, 9083, 9084</div>
|
||
|
||
<div class="related">
|
||
<a href="../combinatorics/combinations-permutations.html">Combinations and Permutations</a>
|
||
<a href="gamma-function.html">Gamma Function</a>
|
||
<a href="index.html">Numbers Index</a>
|
||
</div>
|
||
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|
||
|
||
</article>
|
||
|
||
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|
||
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|
||
<div id="copyrt">Copyright © 2022 Rod Pierce</div>
|
||
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