Combination

The arrangement of a given number of items without the consideration of order. Not to be confused with permutations, which is similar but with consideration of order.

Provided:

• the number of distinct items to choose, $n$ (things you have); and
• the number of combinations, $r$ (‘slots’ to fill)

$${}^n{C}_r=\frac{{}^n{P}_r}{r!}=\frac{\frac{n!}{(n-r)!}}{r!}=\frac{n!}{r!(n-r)!}$$

Where restrictions are applicable, it is important to deal with them first. Some common restrictions include:

• the use of binary genders in a group of people.

Examples

• How many ways are there to select 2 students from a class of 20 students?
  • 2 students, 20 students: no detail of order needed
• How many ways are there to choose 5 students from a class of 20 students to participate in a gaming competition?
  • 5 students, 20 students: no detail of order needed