Algebra
Draft for Information Only ContentTotal Number of Possible Arrangement Total Number of Possible ArrangementPascal's FormulaPascal's Formula, also called Pascal's Rule is a combinatiorial identity.
Binomial ExpansionBinomial Expansion is an algebraic expressions of two terms.
Pascal's TrianglePascal's triangle is a triangular array of the binomial coefficients. Multinomial ExpansionBinomials are just a special case of a larger class of expressions called multinomials expressions with more than one term. The expresion (𝑎+𝑏+𝑐) is a trinomial. Number of Subsets of a SetFor a 10 elements set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, there are 2^10 possible sets. In mathematics of sets, This is not a proper subset of the original set, because it contains the entire set. All other subsets, including the empty set, are considered proper subsets. Therefore, there are 2^10−1 proper subsets of {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. More generally, there are 2^𝑛 subsets of an 𝑛elemnet set, and 2^𝑛1 proper subsets of that 𝑛nelement set.Total Possible ArrangementsExample of a binomial birthorder arrangements. For a family with 𝑛 children, 𝑟 of them sons, there are 𝐶(𝑛,𝑟) different birth order arrangements.
The problem of total possible arrangements can be solved by relating the combinatorial representation to the binomial expransion.
Two equations can be equated by letting 𝑎 and 𝑏 both equal 1,
Therefore Total Possible ArrangementsThe total number of possible ways to arrange 𝑛 objects with first type of object from 0 to 𝑛 and second type of object from 𝑛 to 0 is
Or
Total Possible ArrangementsGiven 𝑛 objects of 𝑟 different types, then there are 𝑟^{𝑛} total possible ways to arrange the 𝑛 objects given all possible ways to group the 𝑟 types.
Finite DifferencesPascal's Triangle ©sideway References
ID: 190500011 Last Updated: 2019/5/11 Revision: Ref: 
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