Combinations

INFO

Suppose a set $S$ with $n$ distinct elements and you want to know how many ways to form a subset of $k$ different elements of $S$.

$$C(n,k)=\frac{P(n,k)}{k!}=\frac{n!}{k!(n-k)!}$$

IMPORTANT

In combinations, order does not matter.

See more in Set Theory

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Examples

1. Choosing 7 People for a Field Trip

How many ways can I choose 7 people from a class of 50?

$$C(50,7)$$

2. Assigning 7 People to 7 Days

If each of the 7 selected people is assigned to a different day of the week (i.e., order matters):

$$P(50,7)$$

3. Assigning Roles in a Group Project

From a team of 20 people, how many ways can you assign 4 distinct roles (leader, scribe, treasurer, presenter)?

$$P(20,4)=20\cdot 19\cdot 18\cdot 17$$

4. Selecting 4 People to Attend a Conference

From a team of 20 people, how many ways can you choose 4 to attend a conference (no roles, just selection)?

$$C(20,4)=\frac{20\cdot 19\cdot 18\cdot 17}{4\cdot 3\cdot 2\cdot 1}$$

NOTE

Combinations are often contrasted with permutations. See rPermutations for ordered selections.