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The Binomial Theorem Revisited

Look familiar?

Dude!  It's Pascal's Triangle?

Those "choose" things make the numbers in Pascal's Triangle!  I think that's pretty dang cool!

What did we use Pascal's Triangle for?

The coefficients of binomial expansions!

So, we can be a LOT more efficient.

Let's crunch an example and you'll see how it works:

( x + y )^( 3 ) = ( 3 0 ) x^( 3 ) y^( 0 ) + ( 3 1 ) x^( 2 ) y^( 1 ) + ( 3 2 ) x^( 1 ) y^( 2 ) + ( 3 3 ) x^( 0 ) y^( 3 )
 

It's even easier to see what's going on when we write out the coefficients:

= ( 3! / 3! 0! ) x^( 3 ) y^( 0 ) + ( 3! / 2! 1! ) x^( 2 ) y^( 1 ) + ( 3! / 1! 2! ) x^( 1 ) y^( 2 ) + ( 3! / 0! 3! ) x^( 0 ) y^( 3 ) ... look at the bottom of the fractions and look at the exponents!
 

= x^( 3 ) + 3x^( 2 )y + 3xy^( 2 ) + y^( 3 )
 

Here's the form for ONE of these guys:

For the expansion of   ( a + b )^( n )

( n! / ( n - r )! r! ) * a^( n - r ) * b^( r )