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

1 Answer
Dec 15, 2015

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.

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