**Uses**- It is used in statistics to calculate the binomial distribution.(Probability distribution is the probability distribution of the number of successes in a sequence of n independent in yes/no distribution)

- This allows statisticians to determine the probability of a given number of favorable outcomes in a repeated number of trials.

- Binomial expansion is also interesting from a mathematical point of view--it gives mathematicians insight into the properties of polynomials.

- Used in the distribution of IP addresses

**Conceptual aspects**-This is the formula.

The way to derive the expanded form of a binomial with a exponential e.g. 4.

Lets just have the binomial be (a+b)

(a+b)^3 = a^3 + 3(a^2)b + 3ab^2 + b^3

As can be seen, the a exponents go from 3 to 0. (a^3, 3(a^2)b, 3ab^2, b^3)

The b exponents go from 0 to 3. (a^3, 3(a^2)b, 3ab^2, b^3)

The coefficients go 1,3,3,1 (3rd row of the Pascal triangle)

To get the coefficients, you can use the above formula. Factorial = !

k is the order of the terms (e.g. 2nd or 3rd term)

e.g. to get the coefficient of the 42nd term of a binomial to the power of 50 e.g. ( a+b )^50

50! / (42!)(50-42)! = 536878650

You can get the coefficient of any term using this formula. Used together with the pattern of the exponents of the a and b terms, you can get the expanded form. ( faster than expanding). It is still very slow.

Below is an example of how to use the formula for binomial expansion to get the expanded form of the binomial cubed.

**How is it used in probability?**
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