Counting Principles
Counting Principles
Counting principles are methods used to find the number of possible outcomes for a given situation. The fundamental counting principle states that, for a series of independent events, the number of outcomes can be found by multiplying the number of possible outcomes for each event. For example, if a die is rolled (6 possible outcomes) and a coin is tossed (2 possible outcomes), there are 6 × 2 = 12 total possible outcomes.
Combinations and permutations describe how many ways a number of objects taken from a group can be arranged. The number of objects in the group is written as n, and the number of objects to be arranged is represented by r (or k).
In a combination, the order of the selections does not matter because every available slot to be filled is the same. Examples of combinations include:
- choosing 3 people from a group of 12 to form a committee (220 possible committees)
- choosing 3 pizza toppings from 10 options (120 possible pizzas)
In a permutation, the order of the selection matters, meaning each available slot is different. Examples of permutations include:
- awarding gold, silver, and bronze medals in a race with 100 participants (970,200 possibilities)
- selecting a president, vice-president, secretary, and treasurer from a committee of 12 people (11,880 possibilities)
The formulas for both calculations are similar. The only difference—the r! in the denominator of a combination—accounts for redundant outcomes. Note that both permutations and combinations can be written in several different shortened notations.
Permutation: \(P(n,r) = nPr = \frac{n!}{(n\ −\ r)!}\)
Combination: \(C(n,r) = nCr = \frac{n}{r} = \frac{n!}{(n\ −\ r)!r!}\)
The notation n! means to multiply all the whole numbers from the given number n down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120