# Difference between revisions of "Combinatorial identity"

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==Vandermonde's Identity== | ==Vandermonde's Identity== | ||

Vandermonde's Identity states that <math>\sum_{k=0}^r\binom mk\binom n{r-k}=\binom{m+n}r</math>, which can be proven combinatorially by noting that any combination of <math>r</math> objects from a group of <math>m+n</math> objects must have some <math>0\le k\le r</math> objects from group <math>m</math> and the remaining from group <math>n</math>. | Vandermonde's Identity states that <math>\sum_{k=0}^r\binom mk\binom n{r-k}=\binom{m+n}r</math>, which can be proven combinatorially by noting that any combination of <math>r</math> objects from a group of <math>m+n</math> objects must have some <math>0\le k\le r</math> objects from group <math>m</math> and the remaining from group <math>n</math>. | ||

+ | '''''Bold text'''''==Hockey-Stick Identity== | ||

+ | For <math>n,r\in\mathbb{N}, n>r,\sum^n_{i=r}{i\choose r}={n+1\choose r+1}</math>. | ||

+ | |||

+ | <asy> | ||

+ | int chew(int n,int r){ | ||

+ | int res=1; | ||

+ | for(int i=0;i<r;++i){ | ||

+ | res=quotient(res*(n-i),i+1); | ||

+ | } | ||

+ | return res; | ||

+ | } | ||

+ | for(int n=0;n<9;++n){ | ||

+ | for(int i=0;i<=n;++i){ | ||

+ | if((i==2 && n<8)||(i==3 && n==8)){ | ||

+ | if(n==8){label(string(chew(n,i)),(11+n/2-i,-n),p=red+2.5);} | ||

+ | else{label(string(chew(n,i)),(11+n/2-i,-n),p=blue+2);} | ||

+ | } | ||

+ | else{ | ||

+ | label(string(chew(n,i)),(11+n/2-i,-n)); | ||

+ | } | ||

+ | } | ||

+ | } | ||

+ | </asy> | ||

+ | |||

+ | This identity is known as the ''hockey-stick'' identity because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed. | ||

+ | |||

+ | |||

+ | ===Proof=== | ||

+ | |||

+ | '''Inductive Proof''' | ||

+ | |||

+ | This identity can be proven by induction on <math>n</math>. | ||

+ | |||

+ | <u>Base Case</u> | ||

+ | Let <math>n=r</math>. | ||

+ | |||

+ | <math>\sum^n_{i=r}{i\choose r}=\sum^r_{i=r}{i\choose r}={r\choose r}=1={r+1\choose r+1}</math>. | ||

+ | |||

+ | <u>Inductive Step</u> | ||

+ | Suppose, for some <math>k\in\mathbb{N}, k>r</math>, <math>\sum^k_{i=r}{i\choose r}={k+1\choose r+1}</math>. | ||

+ | Then <math>\sum^{k+1}_{i=r}{i\choose r}=\left(\sum^k_{i=r}{i\choose r}\right)+{k+1\choose r}={k+1\choose r+1}+{k+1\choose r}={k+2\choose r+1}</math>. | ||

+ | |||

+ | '''Algebraic Proof''' | ||

+ | |||

+ | It can also be proven algebraically with [[Pascal's Identity]], <math>{n \choose k}={n-1\choose k-1}+{n-1\choose k}</math>. | ||

+ | Note that | ||

+ | |||

+ | <math>{r \choose r}+{r+1 \choose r}+{r+2 \choose r}+\cdots+{r+a \choose r}</math> | ||

+ | <math>={r+1 \choose r+1}+{r+1 \choose r}+{r+2 \choose r}+\cdots+{r+a \choose r}</math> | ||

+ | <math>={r+2 \choose r+1}+{r+2 \choose r}+\cdots+{r+a \choose r}=\cdots={r+a \choose r-1}+{r+a \choose r}={r+a+1 \choose r+1}</math>, which is equivalent to the desired result. | ||

+ | |||

+ | '''Combinatorial Proof''' | ||

+ | |||

+ | Imagine that we are distributing <math>n</math> indistinguishable candies to <math>k</math> distinguishable children. By a direct application of Balls and Urns, there are <math>{n+k-1\choose k-1}</math> ways to do this. Alternatively, we can first give <math>0\le i\le n</math> candies to the oldest child so that we are essentially giving <math>n-i</math> candies to <math>k-1</math> kids and again, with Balls and Urns, <math>{n+k-1\choose k-1}=\sum_{i=0}^n{n+k-2-i\choose k-2}</math>, which simplifies to the desired result. | ||

==Another Identity== | ==Another Identity== |

## Revision as of 13:38, 3 August 2010

## Vandermonde's Identity

Vandermonde's Identity states that , which can be proven combinatorially by noting that any combination of objects from a group of objects must have some objects from group and the remaining from group .
* Bold text*==Hockey-Stick Identity==
For .

This identity is known as the *hockey-stick* identity because, on Pascal's triangle, when the addends represented in the summation and the sum itself are highlighted, a hockey-stick shape is revealed.

### Proof

**Inductive Proof**

This identity can be proven by induction on .

__Base Case__
Let .

.

__Inductive Step__
Suppose, for some , .
Then .

**Algebraic Proof**

It can also be proven algebraically with Pascal's Identity, . Note that

, which is equivalent to the desired result.

**Combinatorial Proof**

Imagine that we are distributing indistinguishable candies to distinguishable children. By a direct application of Balls and Urns, there are ways to do this. Alternatively, we can first give candies to the oldest child so that we are essentially giving candies to kids and again, with Balls and Urns, , which simplifies to the desired result.

## Another Identity

### Hat Proof

We have different hats. We split them into two groups, each with k hats: then we choose hats from the first group and hats from the second group. This may be done in ways. Evidently, to generate all possible choices of hats from the hats, we must choose hats from the first and the remaining hats from the second ; the sum over all such is the number of ways of choosing hats from . Therefore , as desired.