Name ______Date ______
Algebra II & TrigonometryPermutations and Combinations
A2.S.9Differentiate between situations requiring permutations and those requiring combinations
A2.S.10Calculate the number of possible permutations (n P r) of n items taken r at a time
A2.S.11Calculate the number of possible combinations (n C r) of n items taken r at a time
A2.S.12Use permutations, combinations, and the Fundamental Principle of Counting to determine the
number of elements in a sample space and a specific subset (event)
Permutations
A permutation is an arrangement of items in a specific order (that is permutations are used when the order of selection is important).
The notation for a permutation of n items taken r at a time is .
Ex: In how many different orders can the 8 students competing in the 200-meter race cross the finish line?
Solution: Obviously, with a race, order matters.
We want to find the order of all 8 students finishes, so in our permutation, n = 8, r = 8.
= 40,320You can also use 8! for this.
Ex: How many different ways can first, second, and third place be decided among the twelve horses running in the Kentucky Derby?
Solution: First place can be won by 12 horses, second by 11, and third by 10.
1. A restaurant critic decides to sample 6 of the 9 desserts on the menu. In how many different orders can this be accomplished?
(1) 720(3) 241,920
(2) 60,480(4) 362,880
Ex: How many different arrangements of the letters in the word TOMORROW can be made?
Solution: The word has 8 letters, but there are multiple O’s and R’s. We must eliminate the extra arrangements of O’s and R’s. To do this, we use the formula where n is the number of things taken n at a time where r are identical. In this example there are 3 O’s and 2 R’s that are repeated, therefore we must divide by both 3! and 2!.
2. How many different 5-letter arrangements can be made from the letters of the word TOOTH?
(1) 12(3) 30
(2) 24(4) 60
Ex: Given the set of numbers {1, 4, 5, 7, 8}, if each digit can be used only once, how many different
- four-digit numbers can be formed{120}
- four-digit odd numbers can be formed{72}
- three-digit numbers larger than 700 can be formed{24}
Combinations
A selection in which order is not important is called a combination. The notation for a combination is
This is equivalent to a permutation of n items taken r at a time, divided by the number of ways the r items can be arranged (because the order doesn’t matter).
Ex: Twelve students are trying out for the basketball team. If all students are equally skilled, in how many ways can the coach choose five starters?
Solution: How many arrangements of 5 items out of 12 can be made in which order does not matter?
Ex: the local community board consists of 12 men and 9 women. If the county needs a representative committee of 3 people, how many
- committees of 3 can be made?
- Committees of 2 men and 1 woman can be formed?
- Committees of only women can be selected?
Solution:
a. There are 21 board members in all and any 3 can be chosen,
b. There is a choice of 2 men out of 12 and a choice of 1 woman out of 9.
c. If the choice is to be made from only women, we disregard the 12 men and select only from the women.
Questions:
- How many different four-digit odd numbers greater than 7,000 can be made from the digits {1, 3, 4, 6, 8} if each digit can be used only once?
(1)12(2) 24(3) 48(4) 120
- The finalists in the 2008 Westminster Kennel Club Dog Show at Madison Square Garden included a 15-inch beagle, a toy poodle, a Sealyham terrier, an Akita, an Australian shepherd, a standard poodle, and a Weimaraner. Which answer below does not represent the number of different ways in which these dogs might have finished in the Best in Show judging?
(1) (2) (3) 7!(4) 5,040
- A committee of 7 is to be chosen from 15 sophomores to design their class ring. Which of the following is not a formula that could be used to determine in how many ways this committee could be chosen?
(1) (2) (3) (4)
- The student court needs 4 juniors and 5 seniors for its panel. If there are 9 juniors who volunteered and 11 seniors, how many different courts could be created?
(1) 2,002(2) 5,040(3) 11,088(4) 58,212
- The school’s summer reading list offers 8 biographies, 6 nonfiction commentaries on world events, and 12 novels to choose from. If students are required to read 2 biographies, 1 nonfiction commentary, and 2 novels, how many different selections of 5 books are possible?
(1) 720(2) 3,003(3) 5,040(4) 11,088