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| In [[mathematics]], the '''multinomial theorem''' says how to expand a [[power (mathematics)|power]] of a sum in terms of powers of the terms in that sum. It is the generalization of the [[binomial theorem]] to polynomials.
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| ==Theorem==
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| For any positive integer ''m'' and any nonnegative integer ''n'', the multinomial formula tells us how a sum with ''m'' terms expands when raised to an arbitrary power ''n'':
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| :<math>(x_1 + x_2 + \cdots + x_m)^n
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| = \sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m}
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| \prod_{1\le t\le m}x_{t}^{k_{t}}\,,</math>
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| where
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| :<math> {n \choose k_1, k_2, \ldots, k_m}
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| = \frac{n!}{k_1!\, k_2! \cdots k_m!}</math>
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| is a '''multinomial coefficient'''. The sum is taken over all combinations of [[nonnegative]] [[integer]] indices ''k''<sub>1</sub> through ''k''<sub>''m''</sub> such that the sum of all ''k''<sub>i</sub> is ''n''. That is, for each term in the expansion, the exponents of the ''x''<sub>''i''</sub> must add up to ''n''. Also, as with the [[binomial theorem]], quantities of the form ''x''<sup>0</sup> that appear are taken to equal 1 (even when ''x'' equals zero).
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| In the case ''m'' = 2, this statement reduces to that of the binomial theorem.
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| ===Example===
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| The third power of the trinomial ''a'' + ''b'' + ''c'' is given by | |
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| :<math>(a+b+c)^3 = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 b^2 a + 3 b^2 c + 3 c^2 a + 3 c^2 b + 6 a b c.</math>
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| This can be computed by hand using the distributive property of multiplication over addition, but it can also be done (perhaps more easily) with the multinomial theorem, which gives us a simple formula for any coefficient we might want. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example:
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| :<math>a^2 b^0 c^1 </math> has the coefficient <math>{3 \choose 2, 0, 1} = \frac{3!}{2!\cdot 0!\cdot 1!} = \frac{6}{2 \cdot 1 \cdot 1} = 3</math>
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| :<math>a^1 b^1 c^1</math> has the coefficient <math>{3 \choose 1, 1, 1} = \frac{3!}{1!\cdot 1!\cdot 1!} = \frac{6}{1 \cdot 1 \cdot 1} = 6</math>.
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| ===Alternate expression===
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| The statement of the theorem can be written concisely using [[multiindices]]:
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| :<math>(x_1+\cdots+x_m)^n = \sum_{|\alpha|=n}{n \choose \alpha}x^\alpha</math>
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| where α = (α<sub>1</sub>,α<sub>2</sub>,…,α<sub>''m''</sub>) and x<sup>α</sup> = ''x''<sub>1</sub><sup>α<sub>1</sub></sup>''x''<sub>2</sub><sup>α<sub>2</sub></sup>⋯''x''<sub>''m''</sub><sup>α<sub>''m''</sub></sup>.
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| ===Proof===
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| This proof of the multinomial theorem uses the [[binomial theorem]] and [[Mathematical induction|induction]] on ''m''.
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| First, for ''m'' = 1, both sides equal ''x''<sub>1</sub><sup>''n''</sup> since there is only one term ''k''<sub>1</sub> = ''n'' in the sum. For the induction step, suppose the multinomial theorem holds for ''m''. Then
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| :<math>(x_1+x_2+\cdots+x_m+x_{m+1})^n = (x_1+x_2+\cdots+(x_m+x_{m+1}))^n </math> | |
| :<math> = \sum_{k_1+k_2+\cdots+k_{m-1}+K=n}{n\choose k_1,k_2,\ldots,k_{m-1},K} x_1^{k_1}x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}(x_m+x_{m+1})^K
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| </math>
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| by the induction hypothesis. Applying the binomial theorem to the last factor,
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| :<math> = \sum_{k_1+k_2+\cdots+k_{m-1}+K=n}{n\choose k_1,k_2,\ldots,k_{m-1},K} x_1^{k_1}x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}\sum_{k_m+k_{m+1}=K}{K\choose k_m,k_{m+1}}x_m^{k_m}x_{m+1}^{k_{m+1}}</math>
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| :<math> = \sum_{k_1+k_2+\cdots+k_{m-1}+k_m+k_{m+1}=n}{n\choose k_1,k_2,\ldots,k_{m-1},k_m,k_{m+1}} x_1^{k_1}x_2^{k_2}\cdots x_{m-1}^{k_{m-1}}x_m^{k_m}x_{m+1}^{k_{m+1}}
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| </math>
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| which completes the induction. The last step follows because
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| :<math>{n\choose k_1,k_2,\ldots,k_{m-1},K}{K\choose k_m,k_{m+1}} = {n\choose k_1,k_2,\ldots,k_{m-1},k_m,k_{m+1}},</math>
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| as can easily be seen by writing the three coefficients using factorials as follows:
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| :<math> \frac{n!}{k_1! k_2! \cdots k_{m-1}!K!} \frac{K!}{k_m! k_{m+1}!}=\frac{n!}{k_1! k_2! \cdots k_{m+1}!}.</math>
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| ==Multinomial coefficients==
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| The numbers
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| :<math> {n \choose k_1, k_2, \ldots, k_m}
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| = \frac{n!}{k_1!\, k_2! \cdots k_m!},</math>
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| which can also be written as
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| :<math>
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| = {k_1\choose k_1}{k_1+k_2\choose k_2}\cdots{k_1+k_2+\cdots+k_m\choose k_m}
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| = \prod_{i=1}^m {\sum_{j=1}^i k_j \choose k_i}</math>
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| are the [[Binomial coefficient#Generalization to multinomials|multinomial coefficients]]. Just like "n choose k" are the coefficients when you raise a ''binomial'' to the ''n''<sup>th</sup> power (e.g. the coefficients are 1,3,3,1 for (''a'' + ''b'')<sup>3</sup>, where ''n'' = 3), the multinomial coefficients appear when one raises a ''multinomial'' to the ''n''<sup>th</sup> power (e.g. (''a'' + ''b'' + ''c'')<sup>3</sup>)
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| ===Sum of all multinomial coefficients===
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| The substitution of ''x''<sub>''i''</sub> = 1 for all ''i'' into:
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| :<math>\sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} x_1^{k_1} x_2^{k_2} \cdots x_m^{k_m}
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| = (x_1 + x_2 + \cdots + x_m)^n\,,</math>
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| gives immediately that
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| :<math>
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| \sum_{k_1+k_2+\cdots+k_m=n} {n \choose k_1, k_2, \ldots, k_m} = m^n\,.
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| </math>
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| ===Number of multinomial coefficients===
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| The number of terms in multinomial sum, #<sub>''n'',''m''</sub>, is equal to the number of monomials of degree ''n'' on the variables ''x''<sub>1</sub>, …, ''x''<sub>''m''</sub>:
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| :<math>
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| \#_{n,m} = {n+m-1 \choose m-1} = {n+m-1 \choose n}\,.
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| </math>
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| The count can be performed easily using the method of [[Stars and bars (combinatorics)|stars and bars]].
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| ===Central multinomial coefficients===
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| All of the multinomial coefficients for which the following holds true:
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| :<math>
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| \left\lfloor\frac{n}{m}\right\rfloor \le k_i \le \left\lceil\frac{n}{m}\right\rceil,\ \sum_{i=1}^m{k_i} = n,
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| </math>
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| are '''central multinomial coefficients''': the greatest ones and all of equal size.
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| A special case for ''m'' = 2 is [[central binomial coefficient]].
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| ==Interpretations==
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| ===Ways to put objects into boxes===
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| The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing ''n'' distinct objects into ''m'' distinct bins, with ''k''<sub>1</sub> objects in the first bin, ''k''<sub>2</sub> objects in the second bin, and so on.<ref>{{cite web |url=http://dlmf.nist.gov/ |title=NIST Digital Library of Mathematical Functions |author=[[National Institute of Standards and Technology]] |date=May 11, 2010 |at=[http://dlmf.nist.gov/26.4 Section 26.4] |accessdate=August 30, 2010}}</ref>
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| ===Number of ways to select according to a distribution===
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| In [[statistical mechanics]] and [[combinatorics]] if one has a number distribution of labels then the multinomial coefficients naturally arise from the binomial coefficients. Given a number distribution {''n''<sub>''i''</sub>} on a set of ''N'' total items, ''n''<sub>''i''</sub> represents the number of items to be given the label ''i''. (In statistical mechanics ''i'' is the label of the energy state.)
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| The number of arrangements is found by
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| *Choosing ''n''<sub>1</sub> of the total ''N'' to be labeled 1. This can be done <math>N\choose n_1</math> ways.
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| *From the remaining ''N'' − ''n''<sub>1</sub> items choose ''n''<sub>2</sub> to label 2. This can be done <math>N-n_1 \choose n_2</math> ways.
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| *From the remaining ''N'' − ''n''<sub>1</sub> − ''n''<sub>2</sub> items choose ''n''<sub>3</sub> to label 3. Again, this can be done <math>N-n_1-n_2 \choose n_3</math> ways.
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| Multiplying the number of choices at each step results in:
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| :<math>{N \choose n_1}{N-n_1\choose n_2}{N-n_1-n_2\choose n_3}...=\frac{N!}{(N-n_1)!n_1!}\frac{(N-n_1)!}{(N-n_1-n_2)!n_2!}\frac{(N-n_1-n_2)!}{(N-n_1-n_2-n_3)!n_3!}....</math>
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| Upon cancellation, we arrive at the formula given in the introduction.
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| ===Number of unique permutations of words===
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| The multinomial coefficient is also the number of distinct ways to [[permutation|permute]] a [[multiset]] of ''n'' elements, and ''k<sub>i</sub>'' are the [[Multiplicity (mathematics)|multiplicities]] of each of the distinct elements. For example, the number of distinct permutations of the letters of the word MISSISSIPPI, which has 1 M, 4 Is, 4 Ss, and 2 Ps is
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| :<math>{11 \choose 1, 4, 4, 2} = \frac{11!}{1!\, 4!\, 4!\, 2!} = 34650.</math>
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| (This is just like saying that there are 11! ways to permute the letters—the common interpretation of [[factorial]] as the number of unique permutations. However, we created duplicate permutations, due to the fact that some letters are the same, and must divide to correct our answer.)
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| ===Generalized Pascal's triangle===
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| One can use the multinomial theorem to generalize [[Pascal's triangle]] or [[Pascal's pyramid]] to [[Pascal's simplex]]. This provides a quick way to generate a lookup table for multinomial coefficients.
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| The case of ''n'' = 3 can be easily drawn by hand. The case of ''n'' = 4 can be drawn with effort as a series of growing pyramids.
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| ==See also==
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| * [[Multinomial distribution]]
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| * [[Stars and bars (combinatorics)]]
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| ==References==
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| {{Reflist}}
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| ==External links ==
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| * <code>mutinom.m</code> function in [http://octave.sourceforge.net/specfun/ Specfun] (since 1.1.0) package of [http://octave.sourceforge.net/index.html Octave-Forge] for [[GNU Octave]]. [http://octave.svn.sf.net/viewvc/octave/trunk/octave-forge/main/specfun/inst/multinom.m SVN version]
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| * {{springer|title=Multinomial coefficient|id=p/m065320}}
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| {{DEFAULTSORT:Multinomial Theorem}}
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| [[Category:Factorial and binomial topics]]
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| [[Category:Articles containing proofs]]
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| [[Category:Theorems in algebra]]
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