 A binomial is the sum of two algebraic terms, x plus 3, y plus z cubed, and x minus 5. Well, remember, mathematicians include subtraction with additions. We can find the power of a binomial using algebra. So x plus 3 to the third power, well that's the product of three factors of x plus 3, and we can expand. But for high powers this can be tedious. A plus 2 to the fifth will be the product of five factors of A plus 2, which will be a lot of work. So let's consider a venue approach. Consider an expansion like this. To find any term in the expansion, we choose one of the terms in each factor and multiply them together. If we select A every time, we get the term A to the fifth. If we select A, A to, A and 2, we get 4A cubed. If we select A, A to, 2 to, we get 8A squared. If we select A to, 2 to, A, A, we get 4A cubed again. So every product will appear once for every time we can select its factors. So we can find the terms in A plus 2 to the fifth power. A key strategy, organize. Since we can choose an A or a 2 in each factor, let's organize by the number of times we choose A and assume we choose 2 the remaining times. So our expansion is, if we pick A five times and multiply we get, which is one term in our expansion, and we can do this in 5 choose 5 one way, so there's only one A to the fifth term. If we pick A four times, we must pick 2 once, and this gives us, we can pick A four times in 5 choose 4 five ways, so there are five such terms giving us. If we pick A three times, we must pick 2 twice giving us, and there are 5 choose 3 ten ways to do this, so our expansion will include 10 summands for A cubed or 4A cubed. If we pick A two times, we must pick 2 three times giving us, and this term will appear 5 choose 2 ten times giving us 80 A squared. If we pick A once, we must pick 2 four times giving us, and this term will appear 5 choose 1 five times giving us, and finally we can pick A zero times, in which case we must pick 2 five times giving us, and this appears 5 choose 0 one time. So the proceeding gives us the expansion. Now in math and in life, aesthetics count. So while we could state our result that way, we prefer to count up. So let's think about this. Remember we're choosing 5 factors. Choosing A five times is the same as choosing 2 zero times. So 5 choose 5 is the same as 5 choose 0. Likewise, choosing A four times is the same as choosing 2 one time. So 5 choose 4 is the same as 5 choose 1. And we also have 5 choose 3 being the same as 5 choose 2. And so we can restate our result, and this gives us our binomial expansion. And the key features in this expansion are that the sum of the exponents is n, the power on the binomial, and a coefficient is n choose i, which we can view as the power on one of the factors. Now the binomial theorem is usually introduced in algebra, which is kind of odd because the least important part of the binomial theorem is its application to algebraic expansions. The most important part of the binomial theorem is what it can tell us about the competitorics, and the key is massage the terms. We'll take a look at what that means next.