Math 2.0

Combinatorics Theorem

Number of k-element subsets

Builds on Cardinality of a finite setCombinationBinomial coefficient

Statement

Let XX be a finite set with ∣X∣=n|X| = n and let kk be a natural number. Then the number of subsets of XX with exactly kk elements is the binomial coefficient (nk)\binom{n}{k}.

Remarks

A proof by induction on nn fixes an element x∈Xx \in X. A kk-subset either omits xx or contains xx, and this gives Pascal's rule. For k>nk > n both sides are 00.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

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Binomial coefficient, definition, no evidence levelBinomial coefficientdefinitionCardinality of a finite set, definition, no evidence levelCardinality of a fini…definitionCombination, definition, no evidence levelCombinationdefinitionNumber of k-element subsets, statement, no evidence levelNumber of k-element s…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binomial coefficient definition
  2. Cardinality of a finite set definition
  3. Combination definition
  4. Number of k-element subsets statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Combination, Number of k-combinations
About this page
Tag0BFMR2FQ8B
URIm2://0BFMR2FQ8B
Kindstatement, theorem
WritingW0 stub
MSC05A10
StatusThis copy of the site has no build data. Run m2 status 0BFMR2FQ8B.
UsesCardinality of a finite set, Combination, Binomial coefficient

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