Binomial Theorem Proof
Definition
- A combination is a subset of a finite set.
- The number of subsets with cardinality from a set with cardinality is denoted by or , and is called the binomial coefficient.
Theorem
Binomial Theorem
Pascal’s Identity
Binomial Coefficient Sum Formula
Binomial Coefficient Subtraction Formula
Binomial Coefficient Squared Sum Formula
Explanation
Note that, except for the Binomial Theorem, the names of the other formulas are not the actual commonly used names, but arbitrary ones created for convenience.
The Binomial Theorem is the most famous and important theorem in combinatorics and is widely applied across various fields.
Proof
The proofs other than the Binomial Theorem are discussed in detail in the respective documents for each formula.
When expanding , the coefficients of are the same as selecting from each , with chosen, and chosen times. Therefore, the combination count becomes the coefficient of , so the following holds true.
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