Question

How do you use the binomial series to expand `(1 + x)^12`?

Answer 1

`1 + 12x + 66x^2 + 220x^3 + 495x^4 + 792x^5 + 924x^6 + 792x^7 + 495x^8 + 220x^9 + 66x^10 +12x^11 + x^12`

Explanation:

The Binomial Theorem is a very useful tool that gives us a way to find the coefficients of binomials raised to some power `n`. In this case, `n=12`, and the theorem says we can find the coefficients one of three ways:

1. Pascal's Triangle: This is the easiest way, but also only useful if you have a copy of the triangle on hand. If you do, look at row 12 (remember the top is row 0) and use the coefficients from that row.

2. The Choose Function: This method actually comes from the theorem itself. The function basically says how many ways there are to choose `r` items from a group of `n` objects where order does not matter.
   Turns out this function can predict binomial coefficients too. You can find them using the `n"C"r` function on your calculator where `n=12` and `r` is increasing from `0 -12`.

3. Factorials: Should your calculator not have an `n"C"r` function, you can evaluate the function directly using factorials (`!`). The factorial of a whole number is that number multiply by all other whole numbers between `1` and itself. The formula for the choose function is:
   `n"C"r = (n!)/(r!(n-r)!)` where `r` and `n` have the same meaning as before.

No matter how you get the coefficients, the answer will still be the same:

`1 + 12x + 66x^2 + 220x^3 + 495x^4 + 792x^5 + 924x^6 + 792x^7 + 495x^8 + 220x^9 + 66x^10 +12x^11 + x^12`

Notice how as the power of `1` decreases (which doesn't actually do anything because `1^n = 1`), the power of `x` increases. The power of `1` is actually the value of `r` for a given coefficient. If you used the power of `x`, then the value of `r` would be `n` minus the exponent.