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

See also Bijective proof.

Algebraic proof

We need to show

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

Let us begin by writing the left-hand side as

${\frac {n!}{k!(n-k)!}}+{\frac {n!}{(k-1)!(n-(k-1))!}}.$

Getting a common denominator and simplifying, we have

${\begin{aligned}&{}\qquad {\frac {n!}{k!(n-k)!}}+{\frac {n!}{(k-1)!(n-k+1)!}}\\\\&={\frac {(n-k+1)n!}{(n-k+1)k!(n-k)!}}+{\frac {kn!}{k(k-1)!(n-k+1)!}}\\\\&={\frac {(n-k+1)n!+kn!}{k!(n-k+1)!}}\\\\&={\frac {(n+1)n!}{k!((n+1)-k)!}}\\\\&={\frac {(n+1)!}{k!((n+1)-k)!}}\\\\&={n+1 \choose k}.\end{aligned}}$

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

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

See also

• Pascal's triangle

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.