Binomial Theorem - Concept
In Algebra II, the binomial theorem describes the explanation of powers of a binomial. When expanding a binomial, the coefficients in the resulting expression are known as binomial coefficients and are the same as the numbers in Pascal's triangle. By using the binomial theorem and determining the resulting coefficients, we can easily raise a polynomial to a certain power.
So factorials are a different way of writing a product. Okay? A series of multiplications. And factorial's basically a exclamation point. So whenever you see something exclamation point, you don't have to yell it out you just have to multiply it by everything below it.
So, n factorial is just that number times a number one less than that times the one number number less than that so on so forth all the way down to the number 1. Okay? So let's take a look at an example.
If I gave you 5 factorial, okay? We start at the number 5 and we just go down multiplying 5 times 4 times 3 times 2 times 1, the one doesn't really matter because anything times 1 is just itself but we always just throw it in there just to be sure we have it. So multiplying this out 5 times 4 is 20 times 3 is 60, times 2 is 120. So 5 factorial is 120. Okay.
We can also have expressions with more than 1 factorial. So say we look at 6 factorial over 4 factorial, 2 factorial. Okay? 6 factorial tells us we are looking at 6 times 5 times 4 times 3 times 2 times 1. 4 factorial tells us we're looking at 4 times 3 times 2 times 1 and 2 factorial is just 2 times 1. Okay.
Now we could multiply this all out but if you look at it a little bit closer, we can actually cancel a lot of things, okay? So 6 factorial is 6 and down and 4 factorial is 4 and down. So those actually share 4, 3, 2 and 1. So right there we know we can cancel out 4, 3, 2 and 1 from the 4 and the 6. Leaving us with the 6 and the 5 in the top, the 2 in the bottom. Again we can cancel like terms, not like terms, what we can factor okay. Take out the contractor. 2 goes into 6. So we're left with 3 times 5 in the top, nothing in the bottom. So this expression is just equal to 15.
So definition of factorial, just taking that number, multiplying it by everything and down and easier way is to simplify it out.