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 →
Temperal (talk | contribs)
3,154 edits
m nav
m ball and urn
Line 22: Line 22:
*Multinomial Coefficients: The number of ways of ordering <math>n</math> objects when <math>r_1</math> of them are of one type, <math>r_2</math> of them are of a second type, ... and <math>r_s</math> of them of another type is <math>\frac{n!}{r_1!r_2!...r_s!}</math>
*Multinomial Coefficients: The number of ways of ordering <math>n</math> objects when <math>r_1</math> of them are of one type, <math>r_2</math> of them are of a second type, ... and <math>r_s</math> of them of another type is <math>\frac{n!}{r_1!r_2!...r_s!}</math>
*Multinomial Theorem: <math>(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</math>. The summation is taken over all sums <math>\sum_{i=1}^{s}r_i</math> so that <math>\sum_{i=1}^{s}r_i=n</math>.
*Multinomial Theorem: <math>(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</math>. The summation is taken over all sums <math>\sum_{i=1}^{s}r_i</math> so that <math>\sum_{i=1}^{s}r_i=n</math>.
===Ball and Urn===
The ball and urn argument states that, there are this many ways to place <math>k</math> balls in <math>n</math> urns:
<math>{n+k-1\choose n-1}</math>


[[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 />
|}<br /><br />

Revision as of 20:22, 10 October 2007



The Problem Solver's Resource
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=17826"