Combinatorics is the branch of mathematics that deals with the number of ways to combine different elements.

Combinatorics can be used as a tool for probability calculations. The combinations found in combinatorics can be used as outcomes in probability calculations.

$P\left(\mathrm{Event}\right)=\frac{\mathrm{favorable}\mathrm{combinations}}{\mathrm{possible}\mathrm{combinations}}$ P(Event) = (favorable combinations) / (possible combinations)

Concepts

“Either Or” (Addition Principle)

If something is "either or," you add the numbers.

Example: You have two bowls with balls. In one bowl, there are 2 balls (one black and one white). In the other bowl, there are 3 balls (one green, one blue, and one red). How many combinations do you have if you either take a ball from bowl 1 or from bowl 2?

Solution: 2 + 3 = 5 possibilities

“Both And” (Multiplication Principle)

If something is "both and," you multiply the numbers.

Example: You have two bowls with balls. In one bowl, there are 2 balls (one black and one white). In the other bowl, there are 3 balls (one green, one blue, and one red). How many combinations do you have if you both take a ball from bowl 1 and a ball from bowl 2?

Solution: 2 * 3 = 6 possibilities

Diagram:

Both the calculation and the tree diagram show 6 possible combinations.

Formulas

Addition: $a+b=\mathrm{total\; combinations}$ (Either or)

Multiplication: $a·b=\mathrm{total\; combinations}$ (Both and)

n = Number of available elements
r = Number of selected elements

Ordered with replacement: $n^r=\mathrm{total\; combinations}$

Ordered without replacement: $\frac{n!}{(n-r)!}$

Unordered with replacement: $\frac{(n+r-1)!}{r!·(n-1)!}$

Unordered without replacement: $\frac{n!}{r!·(n-r)!}$