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pascal behavior (a+b)6

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2 Answers

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 nth 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 = a2 +2ab+b2

That's because when we FOIL out (a+b)2=(a+b)(a+b), we get a2+2ab+b2. 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 = a3 +3a2b +3ab2 + b3

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 nth 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 = a2+2ab+b2 -- 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 a2) and goes up by 1 each time until it gets to 2

(a+b)3 = a3+3a2b+3ab2+b3 -- notice again that the power of 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 a3) 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 6th 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 = a6+6a5b+15a4b2+20a3b3+15a2b4+6ab5+b6


n = 1: 1

n = 2: 1 2 1

n = 3: 1 3 3 1

n = 4: 1 4 6 4 1

n = 5: 1 5 10 10 5 1

n = 6: 1 6 15 20 15 6 1 <==Answer


Attn: Pascal behavior is the way you determine the coefficients of  binomial expansion (a+b)^n by adding the two coefficients from the above row. For example, for the expansion of (a+b)^7, you start with 1, and add the coeffcients from n = 6: 1 6 15 20 15 6 1. You get 1, 1+6, 6+15, 15+20, 20+15, 15+6, 6+1, 1

So, you have

n = 7: 1 7 21 35 35 21 7 1

Hope it helps you!