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The Binomial Theorem

Man!  I'm sick of this!  Watch me start popping these things out by magic:

( a + b)^( 5 ) = a^( 5 ) + 5a^( 4 )b + 10a^( 3 )b^( 2 ) + 10a^( 2 )b^( 3 ) + 5ab^( 4 ) + b^( 5 )

 

( a + b )^( 6 ) = a^( 6 ) + 6a^( 5 )b + 15a^( 4 )b^( 2 ) + 20a^( 3 )b^( 3 ) + 15a^( 2 )b^( 4 ) + 6ab^( 5 ) + b^( 6 )

 

See how I'm doing it?

Four things are going on:

1 )The a guys start at the original power and step down one each time...

                             a^( 6 ) , a^( 5 ) , a^( 4 ) , a^( 3 ) , a^( 2 ) , a^( 1 ) , a^( 0 ) ... no a

2 )The b guys do the opposite...

           no b ... b^( 0 ) , b^( 1 ) , b^( 2 ) , b^( 3 ) , b^( 4 ) , b^( 5 ) , b^( 6 )

3 )The powers on all the ab sets add up to the original power...

      a^( 6 ) , a^( 5 )b^( 1 ) , a^( 4 )b^( 2 ) , a^( 3 )b^( 3 ) , a^( 2 )b^( 4 ) , a^( 1 )b^( 5 ) , b^( 6 ) ... all the powers on each set add up to 6

4 )What about the coefficients?

1 , 6 , 15 , 20 , 15 , 6 , 1

What's going on there?