Pascal's rule

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Not to be confused with Pascal's law.

In mathematics, Pascal's rule is a combinatorial identity about binomial coefficients. It states that for any natural number n we have

where is a binomial coefficient. This is also commonly written

Combinatorial proof[edit]

Pascal's rule has an intuitive combinatorial meaning. Recall that counts in how many ways can we pick a subset with b elements out from a set with a elements. Therefore, the right side of the identity is counting how many ways can we get a k-subset out from a set with n elements.

Now, suppose you distinguish a particular element 'X' from the set with n elements. Thus, every time you choose k elements to form a subset there are two possibilities: X belongs to the chosen subset or not.

If X is in the subset, you only really need to choose k − 1 more objects (since it is known that X will be in the subset) out from the remaining n − 1 objects. This can be accomplished in ways.

When X is not in the subset, you need to choose all the k elements in the subset from the n − 1 objects that are not X. This can be done in ways.

We conclude that the numbers of ways to get a k-subset from the n-set, which we know is , is also the number

See also Bijective proof.

Algebraic proof[edit]

We need to show


Generalization[edit]

Let and . Then

See also[edit]

Sources[edit]

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