Permutation with repetition

In Mathematical, the permutations with repetition of objects of which some are undifferentiated are the various ordered groupings of all these objects. For example, 112,121 and 211 per two digits 1 and one digit 2.

When we permute N objects partially discernible and arranged in a certain order, we find in certain cases the same provision. Let us consider N objects from which K only distinct ( K ≤ N ) is placed in a '' N '' - uplet, and suppose that each one of them appears respectively N 1 time, N 2 time,…, N K time with N 1+ N 2+… + N K = N . When identical elements of this N - uplet is permuted, we obtain same N - uplet.
For example, if we want to determine all the Anagramme S of the MATHEMATICAL word, we see that by exchanging the two letters has , the word remains identical, and on the other hand while transposing the letters E and E we obtain a different word.

; Definition:

That is to say E a finished whole of cardinal K ( K ∈ ℕ∗), E = { X 1, X 2,…, X K }. Are N an entirety such as K ⩽ N and N 1, N 2,…, N K of the natural entireties such as

N 1+ N 2 +… + N K = N .
A permutation of N elements of E with N 1, N 2,…, N K repetitions, is a '' N '' - uplet of elements of E in which each element X 1, X 2,…, X K of E appears N 1, N 2,…, N K time.

; Example:

N - uplet

\ begin {matrix} (\ underbrace {x_1, \ ldots, x_1}, & \ underbrace {x_2, \ ldots, x_2}, & \ ldots, & \ underbrace {x_k, \ ldots, x_k}) \ \ {} _ {n_1 \ rm {\, time}} & {} _ {n_2 \ rm {\, time}} & & {} _ {n_k \ rm {\, time}} \ end {matrix}
is a permutation with particular repetition.

; Theorem:

The number p ( N 1, N 2,…, N K ) of permutations of N elements with N 1, N 2,…, N K repetitions is

p (n_1, n_2, \ ldots, n_k) = \ frac {N!}{n_1! .n_2! \ ldots n_k!}.
This number notes usually C_ {N} ^ {n_1, n_2, \ ldots, n_k} .

; Demonstration:

To build a N - uplet correspondent with a combination containing N 1 time X 1, N 2 time X 2,…, N K time X K , it is enough

  • to choose N 1 sites of the X 1, among N 1 + N 2 +… + N K places available,
  • to choose N 2 sites of the X 2, among N 2 +… + N K remaining places,
  • etc
  • to choose N K sites of the X K , among N K remaining places.
On the whole, there is
C_ {n_1+n_2+ \ ldots+n_k} ^ {n_1}. C_ {n_2+ \ ldots+n_k} ^ {n_2} \ ldots C_ {n_k} ^ {n_k} = \ frac {N!}{n_1! n_2! \ ldots n_k!}.

Application

\ begin {matrix} (x_1+x_2+ \ ldots+x_k) ^n= & \ underbrace {(x_1+x_2+ \ ldots+x_k) (x_1+x_2+ \ ldots+x_k) \ ldots (x_1+x_2+ \ ldots+x_k)}\ \ & {} _ {N \ rm {\, time}} \ end {matrix} The development of this product of factors is a sum of products which can be represented by a N - uplet of elements X 1, X 2,…, X k in which for all 1 ⩽ I ⩽ N , a term of the I - ième factor is with the I ème place.

For all 1 ⩽ I ⩽ K , let us note N I the number of times where X I appears in such a N - uplet. We have

N 1 + N 2 +… + N K = N .
The product corresponding to such a N - uplet is form
x_1^ {n_1} .x_2^ {n_2} \ ldots x_k^ {n_k} .
Being given the natural entireties N 1, N 2,…, N K such as N 1 + N 2 +… + N k= N , the number of terms of the form x_1^ {n_1} .x_2^ {n_2} \ ldots x_k^ {n_k} is the number of permutations of N elements with N 1, N 2,…, N K repetitions.

Thus

(x_1+x_2+ \ ldots+x_k) ^n= \ sum_ {n_1+n_2+ \ ldots+n_k=n} \ frac {N!}{n_1! n_2! \ ldots n_k!}x_1^{n_1}x_2^{n_2}\ldots x_k^{n_k}.

Also see

  • Permutation
  • Coefficient multinomial
  • Formula of the multinôme

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