12.2 Permutations and Combinations

Theorem Multiplication Principle of Counting If a task consists of a sequence of choices in which there are p selections for the first choice, q selections for the second choice, r selections for the third choice, and so on, then the task of making these selections can be done in different ways.

If a license plate consists of a letter, then 5 numbers, how many different types of license plates are possible? license plates

A permutation is an ordered arrangement of n distinct objects without repetitions. The symbol P(n, r) represents the number of permutations of n distinct objects, taken r at a time, where r < n.

Theorem Number of Permutations of n Distinct Objects Taken r at a Time The number of different arrangements from selecting r objects from a set of n objects (r < n), in which 1. the n objects are distinct 2. once an object is used, it cannot be repeated 3. order is important is given by the formula

Evaluate: P (10, 3)

All we know about Anna, Lucy, and John that their birthdays are all in different months. If we were to list all the possible ways this could occur, how many would there be? This is an example of a permutation in which 3 birthdays are selected from a possible 12 months and no birthday may repeat itself. So the 3 requirements are satisfied. 1. the 3 objects are distinct 2. once an object is used, it cannot be repeated 3. order is important

There are 1320 possibilities.

A combination is an arrangement, without regard to order, of n distinct objects without repetitions. The symbol C(n, r) represents the number of combinations of n distinct objects taken r at a time, where r < n.

Theorem Number of Combinations of n Distinct Objects Taken r at a Time The number of different arrangements from selecting r objects from a set of n objects (r < n), in which 1. the n objects are distinct 2. once an object is used, it cannot be repeated 3. order is not important is given by the formula

Evaluate: C (10, 3)

How many committees of 3 people can be formed from out of 8 people? 1. the 8 people are distinct 2. once a person is chosen, he/she cannot be chosen again 3. order is not important Combinations of 8 taken 3 at a time:

How many committees of 3 people (chair, secretary, treasurer) can be formed from out of 8 people? 1. the 8 persons are distinct 2. once a person is chosen, he/she cannot be chosen again 3. order IS IMPORTANT Permutations of 8 taken 3 at a time: