Multinomial Coefficient

Calculates the multinomial coefficient `([n], [n_1 n_2 ... n_k]) \text{ }(n = n_1+n_2+...+n_k)`

Multinomial Coefficient Formula

Let k be integers denoted by `n_1, n_2,\ldots, n_k` such as `n_1+ n_2+\ldots + n_k = n` then the multinominial coefficient of `n_1,\ldots, n_k` is defined by:

$\dbinom {n} {n_1, n_2,\ldots, n_k} =\dfrac {n!} {n_1! \, n_2! \,\ldots n_k!}$

Example:
The multinomial coefficient of integers 2,3,5 is equal to (2 + 3 + 5 = 10):

$\dbinom {10} {2, 3, 5} =\dfrac {10!} {2! \, 3! \, 5!} = 2520$

What is the multinomial coefficient used for?

Calculation of polynomial coefficients
They allow to calculate the coefficients of a polynomial raised to a power n.

Example: calculate the coefficient of$x*y^2*z^3$ in the expansion of$(x + y + z) ^6$

$\text {coef} (x*y^2*z^3) =\dbinom {6} {1,2,3} =\dfrac {6!} {1! \, 2! \, 3!} = 60$,

We calculate all coefficients instead of expanding $(x + y) ^6$ which can be tedious !

Anagrams of a word
The multinomial coefficients are used in some cases of 'permutation' where certain objects in the set to be rearranged are not differentiated.

Example: what is the number of anagrams of the word TUTU?

We notice that the letters T and U are repeated twice so we can't use the classic formula 4! = 24 because there are permutations that are identical.

To illustrate this, let's call 'temporarily' the first T: T(1) and the second T: T(2). Our starting word becomes T(1)UT(2)U. The simple formula 4! takes into account the 2 words T(1)UT(2)U and T(2)UT(1)U as being 2 distinct words which is not true.

In fact, there are $\dbinom {4} {2,2} =\dfrac {4!}{2! \, 2!}= 6$ anagrams that are: TUTU, TTUU, UTUT, UUTT, UTTU, TUUT.

Similarly, the number of anagrams of the word MISSISSIPI can be calculated using the formula: $\dbinom {10} {1,4,4,2} =\dfrac {10!}{1! \, 4! \, 4! \, 2!}= 34\, 650$

See also

Binomial coefficient
Permutation

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