Relationship between pascal triangle and binomial theorem proof

relationship between pascal triangle and binomial theorem proof

Binomial expansions using Pascal's triangle and factorial notation. Total number of subsets. Pascal's Triangle has embedded in it binomial expansions, but isn't actual binomial expansions. You could say [math](a+b)^{n}. Yes, Pascal's Triangle and The Binomial Theorem isn't particularly exciting. But it can, at least, be enjoyable. We dare you to prove us wrong.

You could go like this, you could go like this, or you could go like that.

Binomial Theorem - Pascal's Triangle - An Introduction to Expanding Binomial Series and Sequences.

And then there's only one way to get to that point right over there. And so let's add a fifth level because this was actually what we care about when we think about something to the fourth power.

relationship between pascal triangle and binomial theorem proof

This is essentially zeroth power-- binomial to zeroth power, first power, second power, third power. So let's go to the fourth power. So how many ways are there to get here?

Well I just have to go all the way straight down along this left side to get here, so there's only one way. There's four ways to get here. I could go like that, I could go like that, I could go like that, and I can go like that. There's six ways to go here. Three ways to get to this place, three ways to get to this place.

Binomial Theorem and Pascal's Triangle

So six ways to get to that and, if you have the time, you could figure that out. There's three plus one-- four ways to get here. And then there's one way to get there.

And now I'm claiming that these are the coefficients when I'm taking something to the-- if I'm taking something to the zeroth power. This is if I'm taking a binomial to the first power, to the second power. Obviously a binomial to the first power, the coefficients on a and b are just one and one. But when you square it, it would be a squared plus two ab plus b squared. If you take the third power, these are the coefficients-- third power.

And to the fourth power, these are the coefficients. So let's write them down.

Pascal's triangle and binomial expansion

The coefficients, I'm claiming, are going to be one, four, six, four, and one. And how do I know what the powers of a and b are going to be? Well I start a, I start this first term, at the highest power: And then I go down from there.

And then for the second term I start at the lowest power, at zero. And then b to first, b squared, b to the third power, and then b to the fourth, and then I just add those terms together. And there you have it. I have just figured out the expansion of a plus b to the fourth power. It's exactly what I just wrote down. This term right over here, a to the fourth, that's what this term is.

One a to the fourth b to the zero: This term right over here is equivalent to this term right over there.

relationship between pascal triangle and binomial theorem proof

And so I guess you see that this gave me an equivalent result. Now an interesting question is 'why did this work?

[Discrete Math 1] Binomial Theorem and Pascal's Triangle

Well, to realize why it works let's just go to these first levels right over here. If I just were to take a plus b to the second power. This is going to be, we've already seen it, this is going to be a plus b times a plus b so let me just write that down: So we have an a, an a. We have a b, and a b. We're going to add these together. And then when you multiply it, you have-- so this is going to be equal to a times a. You get a squared. And that's the only way. That's the only way to get an a squared term.

There's only one way of getting an a squared term. Then you're going to have plus a times b.

Pascal's Triangle, Pascal's Formula, The Binomial Theorem and Binomial Expansions

So-- plus a times b. And then you're going to have plus this b times that a so that's going to be another a times b. Plus b times b which is b squared. Theses coefficients can be obtained by the use of Pascal's Triangle. Pascal's Triangle This triangular array is called Pascal's Triangle. Each row gives the combinatorial numbers, which are the binomial coefficients.

Begin and end each successive row with 1. To construct the intervening numbers, add the two numbers immediately above. To construct the next row, begin it with 1, and add the two numbers immediately above: Again, add the two numbers immediately above: Finish the row with 1. There are instances that the expansion of the binomial is so large that the Pascal's Triangle is not advisable to be used. An easier way to expand a binomial raised to a certain power is through the binomial theorem.

It is finding the solution to the problem of the binomial coefficients without actually multiplying out. The theorem is given as: Which can be expanded as: Remember that is another way of writing combination Binomial Theorem also applies to binomial with literal coefficients.

It is given as: