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P1
Given integers n and r where r <= n, generate
all P(n, r) permutations (corresponding to selecting r out of n distinct objects without repetition)
all n^r permutations (corresponding to selecting r out of n distinct objects with repetition)
in which the order is relevant; and
all C(n, r) combinations (corresponding to selecting r out of n distinct objects without repetition)
all C(r+n-1, r) combinations (corresponding to selecting r out of n distinct objects with repetition)
in which the order is irrelevant.
Using Javascript
input: integers n, r (m, k; or m, n) where 0<=r<=n (0<=k<=m-1; or 1<=n<=m, respectively )
output:
Basic (60%)
(1) results of P(n, r), C(n, r), n^r, C(r+n-1, r) with respect to n, r;
More (40%)
(2) (15%) the first n triangular, harmonic, Fibonacci, Lucas, Catalan numbers
with respect to n;
(3) (5%) Eulerian numbers with respect to m, k for 0<=k<=m-1
(i.e., a_sub(m,0), a_sub(m,1), ... , a_sub(m, m-1) ) and
(4) (5%) Stirling number of the second kind with respect to m, n for 1<=n<=m
(i.e., S_sub(m, 1), S_sub(m, 2), ... , S_sub(m, n) )
(5) (15%) Complete all distributions in the following table
Using C++ Builder (NOT recommanded)
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S. J. Shyu Ver. 0.6 2024/09/08, 0.5 2023/04/25, 2019/03/08
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<title>
Permutations and Combinations by S. J. Shyu Ver. 0.8 2024/09/08
</title>
<style> <!-- The <style> tag is used to define style information for an HTML document.
Inside the <style> element you specify how HTML elements should render in a browser.
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body { <!-- font-family: consolas; -->
font-family: Times;
}
table { text-align: center;
border-collapse: collapse;
}
th, td { padding: 5px;
border: 1px solid #dddddd;
}
input[type="number"]
{ width: 60px;
}
</style>
</head>
<body> <!-- The <body> tag defines the document's body.
The <body> element contains all the contents of an HTML document, such as text, hyperlinks, images, tables, lists, etc.
-->
<table bgcolor="b2c1dd"; style="text-align:left;">
<tr>
<td rowspan="3"><img id="myImage" src="https://hackmd-prod-images.s3-ap-northeast-1.amazonaws.com/uploads/upload_720667b4c2ad286fef51b8dffe835d20.jpg?AWSAccessKeyId=AKIA3XSAAW6AWSKNINWO&Expires=1725793740&Signature=M%2FfYq4E60sxqwDnb%2BWzOLNORr6Y%3D" align="center" width="76" height="54">
</td>
</tr>
<tr>
<td> <b><u>Permutations and Combinations</u></b> <br> <font size="-1" color="4477cc"> by Shyong Jian Shyu, sjshyu@gmail.com, Sept. 8, 2024, Ver.0.8
</td>
</tr>
</table>
<table> <!-- The <table> tag defines an HTML table.
An HTML table consists of the <table> element and one or more <tr>, <th>, and <td> elements.
The <tr> element defines a table row, the <th> element defines a table header, and the <td> element defines a table cell.
-->
<tr bgcolor="cfddcf";>
<th>Formula
</th>
<th><label><i>n</i> = </label>
<input type="number" id="n" value="7" min="0">
<label><i>r</i> = </label>
<input type="number" id="r" value="3" min="0">
</th>
<th><button type="button" onclick="computeAll()">Compute All</button>
</th>
<th>Order relevant
</th>
<th>Repetition
</th>
<th>Memo
</th>
</tr>
<tr>
<td>
<label><i>P</i>(<i>n</i>, <i>r</i>)</label>
</td>
<td>
<!--The <span> tag is used to group inline-elements in a document.
The <span> tag provides no visual change by itself.
The <span> tag provides a way to add a hook to a part of a text or a part of a document.
-->
<span id="resPermutation">
</span>
</td>
<td>
<button type="button" onclick="clickPerm()">Permutation</button>
</td>
<td>
yes
</td>
<td>
no
</td>
<td align="left">
# Permutations of size <i>r</i> from <i>n</i> distinct objects
</td>
</tr>
<tr>
<td>
<label><i>C</i>(<i>n</i>, <i>r</i>)</label>
</td>
<td>
<span id="resCombination"></span>
</td>
<td>
<button type="button" onclick="clickComb()">Combination</button>
</td>
<td>
no
</td>
<td>
no
</td>
<td align="left">
# Selections/Combinations of size <i>r</i> from <i>n</i> distinct objects
</td>
</tr>
<tr>
<td>
<label><i>n<sup>r</sup></i></label>
</td>
<td>
<span id="resArrangement"></span>
</td>
<td>
<button type="button" onclick="clickArra()">Arrangement with repetition</button>
</td>
<td>
yes
</td>
<td>
yes
</td>
<td align="left">
# Arrangements with repetition of size <i>r</i> from <i>n</i> distinct objects
</td>
</tr>
<tr>
<td>
<label><i>C</i>(<i>r</i>+<i>n</i>-1, <i>r</i>)</label>
</td>
<td>
<span id="resSelectrofn"></span>
</td>
<td>
<button type="button" onclick="clickSele()">Selection with repetition</button>
</td>
<td>
no
</td>
<td>
yes
</td>
<td align="left">
# Selections with repetition of size <i>r</i> from <i>n</i> distinct objects
</td>
</tr>
</table>
<!-- The <script> tag is used to define a client-side script (JavaScript).
The <script> element either contains scripting statements, or it points to an external script file through the src attribute.
-->
<script>
// <script type="text/javascript">
function Factorial(n)
{ fact = 1;
for (i=2; i<=n; i++)
fact *= i;
return fact;
}
function Perm(n, r)
{ perm = n;
for (i=n-1; i>n-r; i--)
perm *= i;
return perm;
}
function Comb(n, r)
{ return Perm(n, r) / Factorial(r);
}
function invalid(n, r)
{ if (parseInt(n) < 0 || isNaN(parseInt(n)))
{ document.getElementById("n").focus();
return true;
}
else if (parseInt(r) < 0 || isNaN(parseInt(r)))
{ document.getElementById("r").focus();
return true;
}
return false;
}
function computeAll()
{ document.getElementById("resPermutation").innerHTML = "";
document.getElementById("resCombination").innerHTML = "";
document.getElementById("resArrangement").innerHTML = "";
document.getElementById("resSelectrofn").innerHTML = "";
n = document.getElementById("n").value;
r = document.getElementById("r").value;
if (invalid(n, r)) return;
clickPerm();
clickComb();
clickArra();
clickSele();
}
function clickPerm()
{ document.getElementById("resPermutation").innerHTML = "";
n = document.getElementById("n").value;
r = document.getElementById("r").value;
if (invalid(n, r)) return;
document.getElementById("resPermutation").innerHTML = Perm(n, r);
}
function clickComb()
{ document.getElementById("resCombination").innerHTML = "";
n = document.getElementById("n").value;
r = document.getElementById("r").value;
if (invalid(n, r)) return;
document.getElementById("resCombination").innerHTML = Comb(n, r);
}
function clickArra()
{ document.getElementById("resArrangement").innerHTML = ""
n = document.getElementById("n").value;
r = document.getElementById("r").value;
if (invalid(n, r)) return;
document.getElementById("resArrangement").innerHTML = Math.pow(n, r);
}
function clickSele()
{ document.getElementById("resSelectrofn").innerHTML = "";
n = parseInt(document.getElementById("n").value);
r = parseInt(document.getElementById("r").value);
if (invalid(n, r)) return;
document.getElementById("resSelectrofn").innerHTML = Comb(n + r - 1, r);
}
</script>
</body>
</html>
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