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Combinatorial Identities
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Fundamental Counting Principle
If there are ways to do one thing, and ways to do another, then there are ways to do both.
Binomial Theorem
The expansion of is given by .
Permutation of a Set
The number of ways to arrange distinct objects is (factorial).
Combination of a Set
The number of ways to choose items from items without regard to order is .
Pascal's Identity
for .
Permutations with Repetition
is the number of ways to choose items from items with replacement.
Permutations of Multisets
If a set has objects with indistinguishable objects respectively, the number of distinct permutations is .
Combination with Repetition
The number of combinations of items taken at a time with repetition allowed is .
Stirling's Approximation for Factorials
for large .
Catalan Numbers
is the Catalan number.
Derangement Formula
The number of derangements of items is .
Hockey-Stick Identity
For , .
Vandermonde's Identity
for .
Partition of a Set
The number of ways to divide a set of objects into non-empty subsets is given by Stirling numbers of the second kind, .
Multinomial Theorem
The expansion of is given by , where the sum is taken over all non-negative integer sequences such that .
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