Pascal's rule

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

{n-1 \choose k}+{n-1 \choose k-1}={n \choose k}\quad {\text{for }}1\leq k\leq n

where ${n \choose k}$ is a binomial coefficient. This is also commonly written

{n \choose k}+{n \choose k-1}={n+1 \choose k}\quad {\text{for }}1\leq k\leq n+1

Combinatorial proof

Pascal's rule has an intuitive combinatorial meaning. Recall that ${a \choose b}$ 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 ${n \choose k}$ 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 ${n-1 \choose k-1}$ 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 ${n-1 \choose k}$ ways.

We conclude that the numbers of ways to get a k-subset from the n-set, which we know is ${n \choose k}$ , is also the number ${n-1 \choose k-1}+{n-1 \choose k}.$

Algebraic proof

We need to show

{n \choose k}+{n \choose k-1}={n+1 \choose k}.

{\begin{aligned}{n \choose k}+{n \choose k-1}&={\frac {n!}{k!(n-k)!}}+{\frac {n!}{(k-1)!(n-k+1)!}}\\&=n!\left[{\frac {n+1-k}{k!(n+1-k)!}}+{\frac {k}{k!(n+1-k)!}}\right]\\&={\frac {n!(n+1)}{k!(n+1-k)!}}={\binom {n+1}{k}}\end{aligned}}

Generalization

Let $n,k_{1},k_{2},k_{3},\dots ,k_{p},p\in {\mathbb {N}}^{*}\,\!$ and $n=k_{1}+k_{2}+k_{3}+\cdots +k_{p}\,\!$ . Then

{\begin{aligned}&{}\quad {n-1 \choose k_{1}-1,k_{2},k_{3},\dots ,k_{p}}+{n-1 \choose k_{1},k_{2}-1,k_{3},\dots ,k_{p}}+\cdots +{n-1 \choose k_{1},k_{2},k_{3},\dots ,k_{p}-1}\\&={\frac {(n-1)!}{(k_{1}-1)!k_{2}!k_{3}!\cdots k_{p}!}}+{\frac {(n-1)!}{k_{1}!(k_{2}-1)!k_{3}!\cdots k_{p}!}}+\cdots +{\frac {(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots (k_{p}-1)!}}\\&={\frac {k_{1}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}+{\frac {k_{2}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}+\cdots +{\frac {k_{p}(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={\frac {(k_{1}+k_{2}+\cdots +k_{p})(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}\\&={\frac {n(n-1)!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={\frac {n!}{k_{1}!k_{2}!k_{3}!\cdots k_{p}!}}={n \choose k_{1},k_{2},k_{3},\dots ,k_{p}}.\end{aligned}}

Sources

This article incorporates material from Pascal's rule on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
This article incorporates material from Pascal's rule proof on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Merris, Russell. Combinatorics. John Wiley & Sons. 2003 ISBN 978-0-471-26296-1

External links

"Central binomial coefficient". PlanetMath.
"Binomial coefficient". PlanetMath.
"Pascal's triangle". PlanetMath.

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