Why is Pascal's triangle a powerful tool in solving mathematical problems?

[bomba923]

In Pascal's triangle, each element represents the number of ways you can start from the top and get to it! For example, there are six ways to approach the 6 in the row representing the 4th power. There are ten ways to approach the 10 in the row for 5th power...

WHY?

 

[Timbuqtu]

You know how the triangle of Pascal is constructed: each nummber is the sum of the two numbers above it. But then it is easy to see why this yields the number of ways you can get to that number from the top, because you can get there from the first or from the second number above it. So the number of ways is the number of ways to get to the first number above it plus the number of ways to get to the second number.

 

[bomba923]

Hmm---what would be the analytical solution?
The observation is really good---but how would I write a formal proof of this theorem? (the numbers and the #ways from the top theorem!)

 

[matt grime]

It just screams induction, doesn't it.

 

[bomba923]

Although that was the phrasing of my 2nd question---->that's not what I meant
-Indeed it does scream mathematical induction !

This was actually a question from a small booklet asking me to justify my every move/thought in solving the problem. I solved it--but the link to Pascal's triangle seemed abstract at the time (JUST at the time :shy: )

(The idea was to form a link--but I guess that was answered ALREADY by my first question --now I see!)