Difference between revisions of "User:Temperal/The Problem Solver's Resource5"
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*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>. | ||
− | === | + | ===Balls and Urn=== |
− | The | + | The balls 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> | <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]] |
Revision as of 20:40, 10 January 2009
Introduction | Other Tips and Tricks | Methods of Proof | You are currently viewing page 5. |
Contents
Combinatorics
This section cover combinatorics, and some binomial/multinomial facts.
Permutations
The factorial of a number is or also as ,and is denoted by .
Also, .
The number of ways of arranging distinct objects in a straight line is . This is also known as a permutation, and can be notated
Combinations
The number of ways of choosing objects from a set of objects is , which is notated as either or . (The latter notation is also known as taking the binomial coefficient.
Binomials and Multinomials
- Binomial Theorem:
- Multinomial Coefficients: The number of ways of ordering objects when of them are of one type, of them are of a second type, ... and of them of another type is
- Multinomial Theorem: . The summation is taken over all sums so that .
Balls and Urn
The balls and urn argument states that, there are this many ways to place balls in urns: