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### Patterns in Pascal's Triangle

```
Date: 07/21/97 at 14:24:12
From: Camphung Luong
Subject: Pascal's Triangle patterns

I am working on a project about Pascal's triangle. The point of the
project is to find as many patterns as you can and prove it by
induction. My partner and I have already proved one pattern. I am
working on a pattern that the triangle is symmetric, that it's a
palindrome. I have no clue where to start!  I really need help, Dr.
Math!
```

```
Date: 07/21/97 at 15:41:28
From: Doctor Wilkinson
Subject: Re: Pascal's Triangle patterns

I think if you'll look at how you get from one row of the triangle to
the next, you can see quite clearly that every row is a palindrome.
For example, when you go from row 4 to row 5:

1 4 6 4 1

you get

1 1+4 4+6 6+4 4+1 1

and the sums are always formed from symmetrically located terms of
row 4.

Now, how to make a formal proof out of this?

First you need a statement of what it means to be a palindrome.
If you number the terms in row n

0, 1, 2, ..., n

Then to say it is a palindrome means that

term 0 is the same as term n
term 1 is the same as term n-1
term 2 is the same as term n-2

and so on, which we can express by saying that

term k is the same as term n-k

So what you want to prove is the identity

C(n, k) = C(n, n-k)

To show this by induction, you need to show that if it's true for n,
it is true for n + 1.

And what you have to work with is the "Pascal's triangle identity"

C(n, k) = C(n-1, k) + C(n-1, k-1)

That should be enough for you.

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```

```
Date: 07/22/97 at 16:08:25
From: camphung luong
Subject: Patterns in Pascal's Triangle

Hi, Dr. Math.  Thank you for helping me yesterday.  I found out how to
prove it by induction.

I need to find some more patterns. Can you help me?
Thank you again for helping.
```

```
Date: 07/22/97 at 16:50:34
From: Doctor Wilkinson
Subject: Re: Patterns in Pascal's Triangle

Good work!

Have you tried adding up the numbers in each row?

(like 1 - 4 + 6 - 4 + 1, for example)?

-Doctor Wilkinson,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Discrete Mathematics
High School Probability

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