Permutations and Combinations

Permutations and combinations are used to determine the number of possibleoutcomes for a situation. It is sometimes difficult to know which to use.

A ______is an arrangement of outcomes in which the ______.

Ex/ Examine the number of ways to arrange the letters ABC when order does and does not matter.

PermutationCombination

Notice that the number of combinations is less than the number of permutations.

Ex/ Do the following situations involve a permutation or a combination? Explain.

a.Asupremepizzaincludesfivetoppingsfromachoiceof12possible toppings. How many different supreme pizzas are possible?

b.Thecoachisselectingthebattinglineupforhersoftballteam.Howmany ways can she select 9 players out of the 13 that are on the roster?

Once you know if a permutation or a combination should be used, it is simply a matter of applying the correct formula. In both formulas, n represents the total number of objects you have to choose from and r represents the number of objects you can choose.

Permutation:Combination:

Ex/

1.Theprincipalisselectingacommitteeof5studentsoutofthe8students that applied. How many different ways can the committee be selected?

Is this a permutation or combination?

Formula:

2.Awebsiterequiresa3-digitpasswordconsistingoflettersthatcannotbe repeated. How many passwords are possible?

Is this a permutation or combination?

Formula:

Ex/ Find the value of each expression.

1.9P 6

2.4P 4

3.10 P5

Ex/ Find the value of each expression.

1.10 C4

2.8 C 5

3.9 C 2

Theoretical and Experimental Probability

You can find ______by using the following formula.

______relies on data from repeated trials. You can find experimental probability by using the following formula.

Ex/ You choose a crayon at random from a bag containing 4 green crayons, 1 red crayon, 2 blue crayons, and 5 yellow crayons. What is the probability that your crayon will be blue?

**______**

Probability of Compound Events

To find P(A or B) use the following rules.

•For ______events (events with no outcomes in common), P(A or B) = P(A) + P(B).

•For ______events (events with at least one outcome in common), P(A or B) = P(A) + P(B) − P(A and B).

Ex/ You spin a spinner that has 14 equal-sized sections numbered 1 to 14. What is P(odd number or 14)?

Ex/ You spin a spinner with equal-sized sections numbered 1 to 14. What is P(even or greater than 10)?

To find P(A and B) use the following rules.

•For ______events (events that do not affect each other), P(A and B) = P(A) P(B).

•For ______events (events where one event affects the other event), P(A then B) = P(A) P(B after A).

Ex/ You roll a red number cube and a green number cube. What is P(red even and green 1 or 2)?

Ex/ You choose a marble at random from a bag containing 3 red marbles and2 blue marbles. You pick a second marble without replacing the first. What is P(red then blue)?