Combinatorics - Cool math Algebra Help Lessons - The Binomial Theorem Revisited

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Coolmath Algebra
Combinatorics Lesson 6 - The Binomial Theorem Revisited (page 2 of 4)
---- This algebra lesson explains where the coefficients of the binomial theorem (expansion) come from and shows how to find individual terms.

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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:

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

Here's the form for ONE of these guys:

For the expansion of

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

Continued on the next page

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