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Multinomial theorem SE1: coefficient of x^8 in (1+x^2-x^3)^9
Multinomial theorem
Can you find the coefficient of this term??? | Multinomial Theorem
Transcription
Theorem
For any positive integer m and any non-negative integer n, the multinomial formula describes how a sum with m terms expands when raised to an arbitrary power n:
where
is a multinomial coefficient. The sum is taken over all combinations of nonnegativeinteger indices k1 through km such that the sum of all ki is n. That is, for each term in the expansion, the exponents of the xi must add up to n. Also, as with the binomial theorem, quantities of the form x0 that appear are taken to equal 1 (even when x equals zero).
In the case m = 2, this statement reduces to that of the binomial theorem.
Example
The third power of the trinomial a + b + c is given by
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. It is possible to "read off" the multinomial coefficients from the terms by using the multinomial coefficient formula. For example:
has the coefficient
has the coefficient
Alternate expression
The statement of the theorem can be written concisely using multiindices:
First, for m = 1, both sides equal x1n since there is only one term k1 = n in the sum. For the induction step, suppose the multinomial theorem holds for m. Then
by the induction hypothesis. Applying the binomial theorem to the last factor,
which completes the induction. The last step follows because
as can easily be seen by writing the three coefficients using factorials as follows:
The multinomial coefficients have a direct combinatorial interpretation, as the number of ways of depositing n distinct objects into m distinct bins, with k1 objects in the first bin, k2 objects in the second bin, and so on.[1]
Number of ways to select according to a distribution
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 {ni} on a set of N total items, ni represents the number of items to be given the label i. (In statistical mechanics i is the label of the energy state.)
The number of arrangements is found by
Choosing n1 of the total N to be labeled 1. This can be done ways.
From the remaining N − n1 items choose n2 to label 2. This can be done ways.
From the remaining N − n1 − n2 items choose n3 to label 3. Again, this can be done ways.
Multiplying the number of choices at each step results in:
Cancellation results in the formula given above.
Number of unique permutations of words
The multinomial coefficient
is also the number of distinct ways to permute a multiset of n elements, where ki is the multiplicity of each of the ith element. 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