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Wikipedia / Pascal's rule

In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. The binomial coefficients are the numbers that appear in Pascal's triangle. Pascal's rule states that for positive integers n and k is the binomial coefficient, namely the coefficient of the term in the expansion of . There is no restriction on the relative sizes of and ; in particular, the above identity remains valid when whenever .

for all nonnegative integers n for all integers . In this sense, Pascal's rule is the recurrence relation that defines the binomial coefficients.

Pascal's rule can also be generalized to apply to multinomial coefficients.

Combinatorial proof

Pascal's rule has an intuitive combinatorial meaning, that is clearly expressed in this counting proof.

Proof equals the number of subsets with k elements from a set with n elements. Suppose one particular element is uniquely labeled X in a set with n elements.

To construct a subset of k elements containing X, include X and choose k − 1 elements from the remaining n such subsets.

To construct a subset of k elements not containing X, choose k elements from the remaining n such subsets.

Every subset of k elements either contains X or not. The total number of subsets with k elements in a set of n elements is the sum of the number of subsets containing X and the number of subsets that do not contain X.

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Algebraic proof

. Indeed

is used as the extended definition of the binomial coefficient when z is a complex number, thus the above alternative algebraic proof shows that Pascal's rule holds more generally when n is replaced by any complex number.

Generalization

Pascal's rule can be generalized to multinomial coefficients. For any integer p.

The algebraic derivation for this general case is as follows. Let p

See also

References

Bibliography