A simple construction of the triangle proceeds in the following manner. Then, to construct the elements of following rows, add the two above numbers to find the new value. If either of the above numbers is not present, substitute a zero in its place. Applying these numbers to the binomial expansion, we have:. The theorem is given by the formula:.
The coefficients that appear in the binomial expansion are called binomial coefficients. The formula consists of factorials:. For example, [latex]5! The value of [latex]0! Finally, you may recall that the factorial [latex]n! Therefore, we have:. Chapter 3. Matrices and Determinants. Chapter 4. Permutations and Combinations. Chapter 6. Sequence and Series.
Chapter 7. Limit, Continuity, and Differentiability. Chapter 8. Integral Calculus. Chapter 9. Differential Equations. Chapter Coordinate Geometry. Three Dimensional Geometry. Vector Algebra. Statistics and Probability. Mathematical Reasoning. Mathematical Induction.
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Law Change. Animation and Design Change. Let's just multiply this times a plus b to figure out what it is. I'll do this. Let's see. Let's multiply that times a plus b. I'm just going to multiply it this way. First, I'll multiply b times all of these things. I'll do it in this green color. I'll multiply b times all of this stuff. Now let's multiply a times all this stuff. Now when we add all of these things together, we get, we get a to the 3rd power plus, let's see, we have 1 a squared b plus another, plus 2 more a squared b's.
That's going to be 3a squared b plus 3ab squared. Just taking some of the 3rd power, this already took us a little reasonable amount of time, and so you can imagine how painful it might get to do something like a plus b to the 4th power, or even worse, if you're trying to find a plus b to the 10th power, or to the 20th power. This would take you all day or maybe even longer than that. It would be incredibly, incredibly painful.
That's where the binomial theorem becomes useful. What is the binomial theorem? The binomial theorem tells us, let me write this down, binomial theorem.
Binomial theorem, it tells us that if we have a binomial, and I'll just stick with the a plus b for now, if I have, and I'm going to try to color code this a little bit, if I have the binomial a plus b, a plus b, and I'm going to raise it the nth power, I'm going to raise this to the nth power, the binomial theorem tells us that this is going to be equal to, and the notation is going to look a little bit complicated at first, but then we'll work through an actual example, is going to be equal to the sum from k equals 0, k equals 0 to n, this n and this n are the same number, of I don't want to Now this seems a little bit unwieldy.
Let's just review, remind ourselves what n choose k actually means. If we say n choose k, I'll do the same colors, n choose k, we remember from combinatorics this would be equal to n factorial, n factorial over k factorial, over k factorial times n minus k factorial, n minus k factorial, so n minus k minus k factorial, let me color code this, n minus k factorial.
Let's try to apply this. Let's just start applying it to the thing that started to intimidate us, say, a plus b to the 4th power. Let's figure out what that's going to be. Let's try this. So a, and I'm going to try to keep it color-coded so you know what's going on, a plus b, although it takes me a little bit more time to keep switching colors, but hopefully it's worth it, a plus b.
Let's take that to the 4th power. The binomial theorem tells us this is going to be equal to, and I'm just going to use this exact notation, this is going to be the sum from k equals 0, k equals 0 to 4, to 4 of 4 choose k, 4 choose k, 4 choose Now what is that going to be equal to?
Well, let's just actually just do the sum. This is going to be equal to, so we're going to start at k equals 0, so when k equals 0, it's going to be 4 choose 0, 4 choose 0, times a to the 4 minus 0 power, well, that's just going to be a to the 4th power, times b to the 0 power.
This is what we get when k equals 0. Then to that, we're going to add when k equals 1. Well, now, k is 1b to the 1st power.
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