pascal behavior (a+b)6

Robert's answer didn't really help, I imagine; all he did is list the rows of Pascal's Triangle for you. But how is Pascal's Triangle related to your problem, the problem of
*expanding *(a+b)^{6}? It turns out that the numbers in the *n*^{th} row of Pascal's Triangle tell you the
*coefficients *in a binomial expansion -- and this is a great fact, because it helps us expand something like (a+b)^{6}
*without *having to do it the long way, which would be FOILing it all the way out.

Let's look at a familiar example: you may recall that

(a+b)^{2} = a^{2} +2ab+b^{2}

That's because when we FOIL out (a+b)^{2}=(a+b)(a+b), we get a^{2}+2ab+b^{2}. But notice the coefficients of these numbers: 1, 2, 1. That's just the
*second* row of Pascal's Triangle! The connection isn't accidental.

Here's another example: if you FOIL it out (which will take longer than the previous problem), you'll find

(a+b)^{3} = a^{3} +3a^{2}b +3ab^{2} + b^{3}

Again note the coefficients: 1, 3, 3, 1. That's the *third *row of Pascal's triangle!

What we've observed are just two special cases of a general pattern. To perform the
*binomial expansion *(a+b)^{n}, we're going to need the coefficients from the
*n*^{th} row of Pascal's Triangle. But the coefficients aren't enough for us to be able to write down the answer. We need to know what terms the coefficients are coefficients
*of*. Let's go back to our first two examples and try to see a pattern.

(a+b)^{2} = a^{2}+2ab+b^{2} -- notice that the power of *a* starts at 2 and goes down by 1 each time, and the power of *b* starts at 0 (because there are none of them in a^{2}) and goes up by 1 each time until it gets to 2

(a+b)^{3} = a^{3}+3a^{2}b+3ab^{2}+b^{3} -- notice again that the power of
*a *starts at 3 and goes down by 1 each time, and the power of *b* starts at 0 (because there are none of them in a^{3}) and goes up by 1 each time until it gets to 3

This is a general rule. So, **finally**, we can write down the answer to your original question: what is (a+b)^{6}? Well, we should first get the coefficients from the 6^{th} row of Pascal's Triangle (Robert gave you all the rows up to row 6) -- if you don't know how to generate the rows in Pascal's Triangle, I can explain how. The coefficients are 1, 6, 15, 20, 15, 6, 1. Now we use the other pattern we just observed, that the powers of
*a *decrease from 6 to 0 and the powers of *b *increase from 0 to 6. So the
**final answer **is

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