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35 Permutations, Combinations and Proba-
bility
Thus far we have been able to list the elements of a sample space by drawing a tree diagram. For large sample spaces tree diagrams become very complex to construct. In this section we discuss counting techniques for finding the number of elements of a sample space or an event without having to list them.
Permutations Consider the following problem: In how many ways can 8 horses finish in a race (assuming there are no ties)? We can look at this problem as a decision consisting of 8 steps. The first step is the possibility of a horse to finish first in the race, the second step the horse finishes second, ... , the 8th step the horse finishes 8th in the race. Thus, by the Fundamental Principle of counting there are
8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 40, 320 ways
This problem exhibits an example of an ordered arrangement, that is, the order the objects are arranged is important. Such ordered arrangement is called a permutation. Products such as 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 can be written in a shorthand notation called factoriel. That is, 8 · 7 · 6 · 5 · 4 · 3 · 2 · 1 = 8! (read ”8 factoriel”). In general, we define n factoriel by
n! =
n(n − 1)(n − 2) · · · 3 · 2 · 1 , if n ≥ 1 1 , if n = 0.
where n is a whole number n.
Example 35. Evaluate the following expressions:
(a) 6! (b) 10!7!.
Solution. (a) 6! = 6 · 5 · 4 · 3 · 2 · 1 = 720 (b) 10!7! = 10 ·^97 ··^86 ··^75 ··^64 ··^53 ··^42 ··^31 · 2 ·^1 = 10 · 9 · 8 = 720
Using factoriels we see that the number of permutations of n objects is n!.
Example 35. There are 6! permutations of the 6 letters of the word ”square.” In how many of them is r the second letter?
Solution. Let r be the second letter. Then there are 5 ways to fill the first spot, 4 ways to fill the third, 3 to fill the fourth, and so on. There are 5! such permutations.
Example 35. Five different books are on a shelf. In how many different ways could you arrange them?
Solution. The five books can be arranged in 5 · 4 · 3 · 2 · 1 = 5! = 120 ways.
Counting Permutations We next consider the permutations of a set of objects taken from a larger set. Suppose we have n items. How many ordered arrangements of r items can we form from these n items? The number of permutations is denoted by P (n, r). The n refers to the number of different items and the r refers to the number of them appearing in each arrangement. This is equivalent to finding how many different ordered arrangements of people we can get on r chairs if we have n people to choose from. We proceed as follows. The first chair can be filled by any of the n people; the second by any of the remaining (n − 1) people and so on. The rth chair can be filled by (n − r + 1) people. Hence we easily see that
P (n, r) = n(n − 1)(n − 2)...(n − r + 1) =
n! (n − r)!
Example 35. How many ways can gold, silver, and bronze medals be awarded for a race run by 8 people?
Solution. Using the permuation formula we find P (8, 3) = (^) (8−8!3)! = 336 ways.
Example 35. How many five-digit zip codes can be made where all digits are unique? The possible digits are the numbers 0 through 9.
Problem 35. A combination lock has 40 numbers on it.
(a) How many different three-number combinations can be made? (b) How many different combinations are there if the numbers must be all different?
Problem 35. (a) Miss Murphy wants to seat 12 of her students in a row for a class picture. How many different seating arrangements are there? (b) Seven of Miss Murphy’s students are girls and 5 are boys. In how many different ways can she seat the 7 girls together on the left, and then the 5 boys together on the right?
Problem 35. Using the digits 1, 3, 5, 7, and 9, with no repetitions of the digits, how many
(a) one-digit numbers can be made? (b) two-digit numbers can be made? (c) three-digit numbers can be made? (d) four-digit numbers can be made?
Problem 35. There are five members of the Math Club. In how many ways can the positions of officers, a president and a treasurer, be chosen?
Problem 35. (a) A baseball team has nine players. Find the number of ways the manager can arrange the batting order. (b) Find the number of ways of choosing three initials from the alphabet if none of the letters can be repeated.
Combinations As mentioned above, in a permutation the order of the set of objects or peo- ple is taken into account. However, there are many problems in which we want to know the number of ways in which r objects can be selected from n distinct objects in arbitrary order. For example, when selecting a two- person committee from a club of 10 members the order in the committee is irrelevant. That is choosing Mr A and Ms B in a committee is the same as
choosing Ms B and Mr A. A combination is a group of items in which the order does not make a difference.
Counting Combinations Let C(n, r) denote the number of ways in which r objects can be selected from a set of n distinct objects. Since the number of groups of r elements out of n elements is C(n, r) and each group can be arranged in r! ways then P (n, r) = r!C(n, r). It follows that
C(n, r) =
P (n, r) r!
n! r!(n − r)!
Example 35. How many ways can two slices of pizza be chosen from a plate containing one slice each of pepperoni, sausage, mushroom, and cheese pizza.
Solution. In choosing the slices of pizza, order is not important. This arrangement is a combination. Thus, we need to find C(4, 2) = (^) 2!(44!−2)! = 6. So, there are six ways to choose two slices of pizza from the plate.
Example 35. How many ways are there to select a committee to develop a discrete mathe- matics course at a school if the committee is to consist of 3 faculty members from the mathematics department and 4 from the computer science depart- ment, if there are 9 faculty members of the math department and 11 of the CS department?
Solution. There are C(9, 3) · C(11, 4) = (^) 3!(99!−3)! · (^) 4!(1111!−4)! = 27, 720 ways.
Practice Problems
Problem 35. Compute each of the following: (a) C(7,2) (b) C(8,8) (c) C(25,2)
Problem 35. Find m and n so that C(m, n) = 13
Finding Probabilities Using Combinations and Permutations Combinations can be used in finding probabilities as illustrated in the next example.
Example 35. Given a class of 12 girls and 10 boys.
(a) In how many ways can a committee of five consisting of 3 girls and 2 boys be chosen? (b) What is the probability that a committee of five, chosen at random from the class, consists of three girls and two boys? (c) How many of the possible committees of five have no boys?(i.e. consists only of girls) (d) What is the probability that a committee of five, chosen at random from the class, consists only of girls?
Solution. (a) First note that the order of the children in the committee does not matter. From 12 girls we can choose C(12, 3) different groups of three girls. From the 10 boys we can choose C(10, 2) different groups. Thus, by the Fundamental Principle of Counting the total number of committee is
C(12, 3) · C(10, 2) =
(b) The total number of committees of 5 is C(22, 5) = 26, 334. Using part (a), we find the probability that a committee of five will consist of 3 girls and 2 boys to be C(12, 3) · C(10, 2) C(22, 5)
(c) The number of ways to choose 5 girls from the 12 girls in the class is
C(10, 0) · C(12, 5) = C(12, 5) =
(d) The probability that a committee of five consists only of girls is
C(12, 5) C(22, 5)
Practice Problems
Problem 35. John and Beth are hoping to be selected from their class of 30 as president and vice-president of the Social Committee. If the three-person committee (president, vice-president, and secretary) is selected at random, what is the probability that John and Beth would be president and vice president?
Problem 35. There are 10 boys and 13 girls in Mr. Benson’s fourth-grade class and 12 boys and 11 girls in Mr. Johnson fourth-grade class. A picnic committee of six people is selected at random from the total group of students in both classes.
(a) What is the probability that all the committee members are girls? (b) What is the probability that the committee has three girls and three boys?
Problem 35. A school dance committee of 4 people is selected at random from a group of 6 ninth graders, 11 eighth graders, and 10 seventh graders.
(a) What is the probability that the committee has all seventh graders? (b) What is the probability that the committee has no seventh graders?
Problem 35. In an effort to promote school spirit, Georgetown High School created ID numbers with just the letters G, H, and S. If each letter is used exactly three times,
(a) how many nine-letter ID numbers can be generated? (b) what is the probability that a random ID number starts with GHS?
Problem 35. The license plates in the state of Utah consist of three letters followed by three single-digit numbers.
(a) If Edward’s initials are EAM, what is the probability that his license plate will have his initials on it (in any order)?
