Name _____________________________________________
Algebra II & Trigonometry
Date _______________________________________
Permutations and Combinations
A2.S.9
Differentiate between situations requiring permutations and those requiring combinations
A2.S.10
Calculate the number of possible permutations (n P r) of n items taken r at a time
A2.S.11
Calculate the number of possible combinations (n C r) of n items taken r at a time
A2.S.12
Use 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 n Pr .
n Pr  n  n  1 n  2 n  3   n  r  1
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.
You can also use 8! for this.
8 P8  8  7  6  5  4  3  2  1 = 40,320
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.
12 P3  12  11  10  1,320
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
n!
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
r!
example there are 3 O’s and 2 R’s that are repeated, therefore we must divide by both 3! and 2!.
8!
8  7  6  5  4  3  2 1

 3,360
2!3!
3  2  1 2  1
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
a. four-digit numbers can be formed
{120}
___  ___  ___  ___
b. four-digit odd numbers can be formed
{72}
___  ___  ___  ___
c. 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
n
P
  n C r  n r
r!
r 
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?
12  11  10  9  8
12 P5

 792
12 C 5 
5!
5  4  3  2 1
Ex: the local community board consists of 12 men and 9 women. If the county needs a representative committee of 3
people, how many
a. committees of 3 can be made?
b. Committees of 2 men and 1 woman can be formed?
c. Committees of only women can be selected?
Solution:
P
21  20  19
a. There are 21 board members in all and any 3 can be chosen, 21C 3  21 3 
 1,330
5!
3  2 1
b. There is a choice of 2 men out of 12 and a choice of 1 woman out of 9. 12 C 2 9 C1  594
c. If the choice is to be made from only women, we disregard the 12 men and select only from the women. 9 C 3  84
Questions:
1. 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
2. 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) 7 P1
(2) 7 P7
(3) 7!
(4) 5,040
3. 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?
 15 
(1) 15 C 7
(2) 15 C8
(3) 15 P87
(4)  
8 
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
5. 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