Concept

Combination

A combination is a selection of elements from a set where the order in which they are selected is not important. Therefore, the only important thing is which elements are selected. For example, choosing two different ingredients for a salad from five unique options in the salad bar.

When counting the number of possible combinations, either:

every possibility must be listed and counted or
the Combination Formula can be used.

This formula calculates the number of combinations $C$ that can be done by using $n$ elements of a set to form groups of $r$ elements. Note that the exclamation points in the formula indicate that the factorial of the value should be calculated.

$_{n} C_{r} = \frac{n !}{( n - r ) ! r !}$

In the salad bar example, there were $n = 5$ options provided to make groups of $r = 2 .$

$_{5} C_{2}_{5} C_{2}_{5} C_{2}_{5} C_{2}_{5} C_{2} = \frac{5 !}{( 5 - 2 ) ! 2 !} = \frac{5 !}{3 ! 2 !} = \frac{5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{3 \cdot 2 \cdot 1 \cdot 2 \cdot 1} = \frac{2 0}{2} = 10$

An alternate notation for $_{n} C_{r}$ is $C (r, n) .$