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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

In class we studied the identity \displaystyle\binom{r}{r}+\binom{r+1}{r} +\binom{r+2}{r} + \cdots +\binom{n}{r} = \binom{n+1}{r+1} We also took a glimpse at \displaystyle\binom{r}{0}+\binom{r+1}{1} +\binom{r+2}{2} + \cdots +\binom{n}{n-r} = \binom{n+1}{n-r}. We will now take a closer look at this second identity.

(a) Confirm that the second identity works for n=5, r=2 and for n=7, r=3.

(b) What is the relationship between the first and second identities?

(c) Prove the second identity above algebraically without using what you learned in Part b. (In other words, prove it without the help of the hockey stick identity we studied in class).

(d) Prove the second identity above with a block-walking argument.

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi;

Sorry, I can not read that?!

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

srry, i used latex, i will change it to regular english

Genius is one percent inspiration and ninety-nine percent perspiration

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

In class we studied the identity combination{r}{r}+combination{r+1}{r} +combination{r+2}{r} + ... +combination{n}{r} = combination{n+1}{r+1} We also took a glimpse at combination{r}{0}+combination{r+1}{1} +combination{r+2}{2} +... +combination{n}{n-r} = combination{n+1}{n-r}. We will now take a closer look at this second identity.

(a) Confirm that the second identity works for n=5, r=2 and for n=7, r=3.

(b) What is the relationship between the first and second identities?

(c) Prove the second identity above algebraically without using what you learned in Part b. (In other words, prove it without the help of the hockey stick identity we studied in class).

(d) Prove the second identity above with a block-walking argument.

Genius is one percent inspiration and ninety-nine percent perspiration

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

combination{x}{y} means x combination y

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

I know that.

combination{r}{r}+combination{r+1}{r} +combination{r+2}{r} + ... +combination{n}{r} = combination{n+1}{r+1} We also took a glimpse at combination{r}{0}+combination{r+1}{1} +combination{r+2}{2} +... +combination{n}{n-r} = combination{n+1}{n-r}.

It is totally unreadable on my browser

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

i will change it

Genius is one percent inspiration and ninety-nine percent perspiration

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

In class we studied the identity (r) combination (r) + (r+1) combination (r) +(r+2) combination (r) + ... + (n) combination (r) = (n+1) combination r+1 We also took a glimpse at (r) combination (0) + (r+1) combination (1) +(r+2) combination (2) +... +(n) combination (n-r) = (n+1) combination (n-r). We will now take a closer look at this second identity.

(a) Confirm that the second identity works for n=5, r=2 and for n=7, r=3.

(b) What is the relationship between the first and second identities?

(c) Prove the second identity above algebraically without using what you learned in Part b. (In other words, prove it without the help of the hockey stick identity we studied in class).

(d) Prove the second identity above with a block-walking argument.

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi;

What was your answer for a)?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

i didn't know how to do it

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Hi;

Did you substitute the values in?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

i think i got it

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

Very good!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

never mind i didn't get it

Genius is one percent inspiration and ninety-nine percent perspiration

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

how do you substitute it

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

We also took a glimpse at (r) combination (0) + (r+1) combination (1) +(r+2) combination (2) +... +(n) combination (n-r) = (n+1) combination (n-r).

You want to prove this identity for n=5, r=2 and for n=7, r=3?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**mathstudent2000****Member**- Registered: 2013-07-26
- Posts: 79

yes please, but i don't know how to

Genius is one percent inspiration and ninety-nine percent perspiration

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,237

As near as I can understand it, for n=5, r=2.

We are done.

For n=7, r=3.

We are done.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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