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The Binomial Theorem

Let's write them all out to see if we can find a pattern...

  row 1: 1 , ( a + b )^( 0 ) ... row 2: 1 1 , ( a + b )^( 1 ) ... row 3: 1 2 1 , ( a + b )^( 2 ) ... row 4: 1 3 3 1 , ( a + b )^( 3 ) ... row 5: 1 4 6 4 1 , ( a + b )^( 4 ) ... row 6: 1 5 10 10 5 1 , ( a + b )^( 5 ) ... row 7: 1 6 15 20 15 6 1 , ( a + b )^( 6 )
 

What's the trick?

What's the next line?

            1 ... 1 1 ... 1 2 1 ... 1 3 3 1 ... 1 4 6 4 1 ... 1 5 10 10 5 1 ... 1 6 15 20 15 6 1 ... 1 7 21 35 35 21 7 1 ... This is called Pascal's Triangle.

Ones on the edge...  add the two guys above!

Whoa!
 

So, can you write out the expansion of ( a + b )^( 7 ) ?