Art of Problem Solving
Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

User:Temperal/The Problem Solver's Resource5: Difference between revisions

← Older editNewer edit →
m ball and urn
Temperal (talk | contribs)
3,154 edits
rmv
Line 1: Line 1:
__NOTOC__
{{User:Temperal/testtemplate|page 5}}
<br /><br />
{| style='background:lime;border-width: 5px;border-color: limegreen;border-style: outset;opacity: 0.8;width:840px;height:300px;position:relative;top:10px;'
|+ <span style="background:aqua; border:1px solid black; opacity: 0.6;font-size:30px;position:relative;bottom:8px;border-width: 5px;border-color:blue;border-style: groove;position:absolute;top:50px;right:155px;width:820px;height:40px;padding:5px;">The Problem Solver's Resource</span>
|-
| style="background:lime; border:1px solid black;height:200px;padding:10px;" | {{User:Temperal/testtemplate|page 5}}
==<span style="font-size:20px; color: blue;">Combinatorics</span>==
==<span style="font-size:20px; color: blue;">Combinatorics</span>==
This section cover combinatorics, and some binomial/multinomial facts.
This section cover combinatorics, and some binomial/multinomial facts.
Line 29: Line 24:


[[User:Temperal/The Problem Solver's Resource4|Back to page 4]] | [[User:Temperal/The Problem Solver's Resource6|Continue to page 6]]
[[User:Temperal/The Problem Solver's Resource4|Back to page 4]] | [[User:Temperal/The Problem Solver's Resource6|Continue to page 6]]
|}<br /><br />

Revision as of 18:18, 10 January 2009


Introduction Other Tips and Tricks Methods of Proof You are currently viewing page 5.

Combinatorics

This section cover combinatorics, and some binomial/multinomial facts.

Permutations

The factorial of a number $n$ is $n(n-1)(n-2)...(1)$ or also as $\prod_{a=0}^{n-1}(n-a)$,and is denoted by $n!$.

Also, $0!=1$.

The number of ways of arranging $n$ distinct objects in a straight line is $n!$. This is also known as a permutation, and can be notated $\,_{n}P_{r}$

Combinations

The number of ways of choosing $n$ objects from a set of $r$ objects is $\frac{n!}{r!(n-r)!}$, which is notated as either $\,_{n}C_{r}$ or $\binom{n}{r}$. (The latter notation is also known as taking the binomial coefficient.

Binomials and Multinomials

  • Binomial Theorem: $(x+y)^n=\sum_{r=0}^{n}x^{n-r}y^r$
  • Multinomial Coefficients: The number of ways of ordering $n$ objects when $r_1$ of them are of one type, $r_2$ of them are of a second type, ... and $r_s$ of them of another type is $\frac{n!}{r_1!r_2!...r_s!}$
  • Multinomial Theorem: $(x_1+x_2+x_3...+x_s)^n=\sum \frac{n!}{r_1!r_2!...r_s!} x_1+x_2+x_3...+x_s$. The summation is taken over all sums $\sum_{i=1}^{s}r_i$ so that $\sum_{i=1}^{s}r_i=n$.

Ball and Urn

The ball and urn argument states that, there are this many ways to place $k$ balls in $n$ urns:

${n+k-1\choose n-1}$

Back to page 4 | Continue to page 6

Retrieved from "https://wiki.aopstest.com/wiki/index.php?title=User:Temperal/The_Problem_Solver%27s_Resource5&oldid=29341"