Draft talk:Information theory/Permutations and combinations

end
In this case, the Rule of Product applies, which implies that there are $$3\times 2 = 6$$ different ways to select the order in which the subscripts can appear. Consider, for example the combination where the first two letters are $R$. :

Draft:Information theory/k-combination proof
$$\text{If unlabeled: }\Omega = 5! = 120\,\text{outcomes}$$

$$\text{If labeled: }\Omega =\frac{(2+3)!}{2!\, 3!}=10\,\text{outcomes}$$

Permutation $$n^r$$ with repetition and

Combinations $$\tfrac{n!}{r!(n-r)!}$$ without repetition and $$\tfrac{(n+r-1)!}{r!(n-1)!}$$  with repetition.

Partial permutation
$$\tfrac{n!}{(n-r)!}$$ partial permutation, where we sort into two bins, but retain the order of only one of them

Optional material lifted from Combination
A k-combination of a set S is a subset of k distinct elements of S. If the set has n elements, the number of k-combinations is equal to the binomial coefficient,
 * $${\displaystyle {\binom {n}{k}}={\frac {n(n-1)\dotsb (n-k+1)}{k(k-1)\dotsb 1}},}$$

which can be written using factorials as $${\displaystyle \textstyle {\frac {n!}{k!(n-k)!}}}$$

The number of k-combinations from a given set S of n elements is can be written as $$\tbinom nk$$ (spoken as "n choose k"). Other notations include $$C(n,k)$$, $$C^n_k$$, $${}_nC_k$$, $${}^nC_k$$, $$C_{n,k}$$, and $$C_n^k$$.<!--

Bayes' Theorem LIT SEARCH

 * Bayes' theorem Proper derivation: No venn diagrams
 * 1) venn V eikosograms
 * 2) Journey to understanding Bayes' theorem (with rigor)
 * 3) Bayes' Theorem using Venn Diagrams
 * 4) Deriving Bayes' theorem (rigorous)
 * 5) Visualizing Bayes' Theorem States equations but uses Venn diagrams to explain

Combinations

 * Combinatorial proof Vague: two types of mathematical proof
 * 1) pitt.edu bonidie cs441 proves this using Rule of Product and the concept of a r-permutations. Note the confusion of terms by how Partial permutations explains it.
 * 2) Combinatorial Arguments
 * 3) Wichita Hammond 321 how many subsets, taken from Math321 ©2018-2019 John Hammond
 * 4) A google search for proof of formula for combinations quickly uncovered a non-pictorial version of this same proof.

Wikipedia

 * Permutation
 * Combination
 * Multinomial theorem, especially w:special:permalink/1009456770
 * Binomial coefficient, especially w:special:permalink/1038252851
 * Rule of product
 * Stars and bars (combinatorics)
 * http://discrete.openmathbooks.org/dmoi2/sec_stars-and-bars.html
 * Twelvefold way See "n-subset of X"