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

Dec 15, 2015

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

#### 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 $n \text{C} r$ function on your calculator where $n = 12$ and $r$ is increasing from $0 - 12$.

3. Factorials: Should your calculator not have an $n \text{C} r$ 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:

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

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.