Discrete Mathematics - Combinatorial Calculator

Combinatorial Calculator

Combinations and Permutations

Programming Assignment 1: (You may use any programming tool. A Javascript program, which I warmly recommend, is given below.)

input: integers n, r (m, k; or m, n) where 0<=r<=n (0<=k<=m-1; or 1<=n<=m, respectively )

output:

(1) results of P(n, r), C(n, r), n^r, C(r+n-1, r) with respect to n, r;

(2) the first n Catalan, triangular, harmonic, Fibonacci, Lucas with respect to n;

(3) 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) 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) )

Bonus: (Implement all computations in the table below)

/* For those who are interested in coding in Javascript, which I suggest you to learn, please refer to the following ~.html. Please download the file via the link below and run it on your PC, NB, cell phone, etc.

https://drive.google.com/file/d/1CbIXqGgJJCFhRY2mjR9-_wTbJRRDnYwz/view?usp=sharing

(Open the file using Notepad++, Sublime or Notepad to get an easy reading format of well indented codes.)

*/

Source Codes in Javascipt

<!-- I put comments here for you guys to learn html & javascript quickly. Refer to https://www.w3schools.com/tags/default.asp for "tags" in html.

S. J. Shyu Ver. 0.6 2024/09/08, 0.5 2023/04/25, 2019/03/08

-->

<!DOCTYPE html> <!--The <!DOCTYPE> declaration must be the very first thing in your HTML document, before the <html> tag.

The <!DOCTYPE> declaration is not an HTML tag; it is an instruction to the web browser about what version of HTML the page is written in.

Tip: Always add the <!DOCTYPE> declaration to your HTML documents, so that the browser knows what type of document to expect.

-->

<html> <!--The <html> tag tells the browser that this is an HTML document.

The <html> tag represents the root of an HTML document.

The <html> tag is the container for all other HTML elements (except for the <!DOCTYPE> tag).

-->

<head> <!--The <head> element is a container for all the head elements.

The <head> element can include a title for the document, scripts, styles, meta information, and more.

-->

<title>

Permutations and Combinations by S. J. Shyu Ver. 0.8 2024/09/08

</title>

Inside the <style> element you specify how HTML elements should render in a browser.

Each HTML document can contain multiple <style> tags.

-->

body {

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.

-->

<tr>

</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.

-->

<th>Formula

</th>

</th>

<th><button type="button" onclick="computeAll()">Compute All</button>

</th>

<th>Order relevant

</th>

<th>Repetition

</th>

<th>Memo

</th>

</tr>

<tr>

<td>

</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>

</td>

<td>

<button type="button" onclick="clickPerm()">Permutation</button>

</td>

<td>

yes

</td>

<td>

no

</td>

# Permutations of size <i>r</i> from <i>n</i> distinct objects

</td>

</tr>

<tr>

<td>

</td>

<td>

</td>

<td>

<button type="button" onclick="clickComb()">Combination</button>

</td>

<td>

no

</td>

<td>

no

</td>

# Selections/Combinations of size <i>r</i> from <i>n</i> distinct objects

</td>

</tr>

<tr>

<td>

</td>

<td>

</td>

<td>

<button type="button" onclick="clickArra()">Arrangement with repetition</button>

</td>

<td>

yes

</td>

<td>

yes

</td>

# Arrangements with repetition of size <i>r</i> from <i>n</i> distinct objects

</td>

</tr>

<tr>

<td>

</td>

<td>

</td>

<td>

<button type="button" onclick="clickSele()">Selection with repetition</button>

</td>

<td>

no

</td>

<td>

yes

</td>

# 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 type="text/javascript">

function Factorial(n)

{ fact = 1;

for (i=2; i<=n; i++)

fact *= i;

return fact;

}

function Perm(n, r)

{ perm = 1;

for (i=n; 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>