Counting Techniques, Permutations and Combinations

Fundamental Counting Principle

The Fundamental Counting Principle (FCP)If one thing can occur in m ways and a second thing can occur in n ways, and a third thing can occur in p ways, and so on, then the sequence of things can occur in

Example 1The Shirt Mart sells shirts in sizes S, M, L, and XL. Each size comes in five colors: red, yellow, white, orange, and blue. The shirts come in short sleeve and long sleeve. How many kinds of shirts are there?

Example 2A plate number is made up of two consonants followed by three nonzero digits followed by a vowel. How many plate numbers are possible ifa. The letters and digits cannot be

repeated in the same plate number?b. The letters and digits can be repeated

in the same plate number?

Example 3Carla is taking a matching test in which he is supposed to match four answers with four questions. In how many different ways can he answer the four questions?

Practice Exercises1. How many ordered pairs of letters are there that use only the letters

A, B, C, D, and E?2. How many different sequences of heads or tails are possible if a coin

is flipped 8 times?3. A model is selecting her outfits purchased 5 blouses, 4 skirts and 3

blazers. How many different new outfits consisting of a blouse, a skirt and a blazer can she create from her new collection?

4. Five different mathematics books and six different grammar books are to be arranged on a shelf. How many possible arrangements can be made if

a. the books on the same subjects are to be arranged together?

b. the books are to be arranged alternately?5. From the word “ALERT,” determine how many letter arrangements are possible given the following conditions:

a. All 5 letters are used without restrictionsb. all vowels and consonants are togetherc. Only three letters are used without repeating any letter.d. Only four letters are used without restrictions.

Permutation

Factorial NotationIn general, if n is a positive integer, then n factorial denoted by n! is the product of all integers less than or equal to n.

n! = n. (n-1).(n-2). … . 2.1

As a special case, we define0! = 1

DefinitionA permutation is the ordered arrangement of distinguishable objects without allowing repetitions among the objects.

The number of permutations of n things taken n at a time is given by

The permutations of n things taken r at a time is given by

Example 1In how many ways can a president and vice-president be chosen from a club with 12 members?

If there are 50 floats in Penagbenga Festival, how many ways can a first-place, a second-place, and a third-place trophy be awarded?

Example 2Find the number of different arrangments of the set of six letters HONESTa. Taken two at a timeb. Taken three at a timec. Taken six at a time.

Example 3Find the number of permutations in each situation.a. A softball coach chooses the first,

second, and third batters for a team of 10 players.

b. Three-digit numbers are formed from the digits 2, 3, 4, and 5, with no digits repeated.

Example 4Five golfers on a team are playing in a tournament. How many different line-up can the coach make?

Example 5Find the number of ways a president, a vice-president, a secretary, and a treasurer can be chosen from among Alvin, Aris, Richard, Ricky, Alma, and Alice.

Permutations of Identical objects

There are instances when the n things to be arranged are not all different. That is, of them are alike, of them are alike, … , of them are alike where + + …+ = n. The number of permutations of these things therefore is

Practice ExercisesIn how many different ways can four people be seated in a row?

Example 1In how many ways may the letters of the word “STATISTICS” be arranged?

Example 2In how many ways may the letters of the word “ASSESSMENT” be arranged?

Circular PermutationsIf n different things are to have a circular arrangement then the number of permutations is equal to

Example In how many ways may seven persons be seated around a circular table?

Circular Permutations with no definite top or bottom

The number of permutations of n different things around a key ring and the like is

ExampleIn how many ways can 3 keys be arranged in a key ring?

More exercises?Visit www.khanacademy.org , join the stat class 2013 and add me as coach (code: Q87N9A) for me to track your progress and give additional points.

Quiz1. In how many ways may the letters of

the word ASSESSMENT be arranged?2. How many license plates can be

manufactured with three letters followed by three digits? No repetition

3. How many distinguishable ways can 4 beads be arranged in a circular bracelet?

4. In how many ways can 9 people be seated in a round table?

5. How many permutations of letters in the word GOOGOLPLEX?

Combinations

CombinationThe number of combinations of n things taken r at a time is given by

Example 1Ellen received an offer to join a CD club. If she agrees to be a member, she can select 5 CDs from a list of 40 CDs. In how many ways can Ellen select the 5 CDs?

Example 2Mr. Elton has to choose three of the six officers of the Math Club to go to a regional meeting. How many possible choices does he have?

Example 3How many subcommittees of 5 people can be formed from a committee consisting of 8 people?

Example 4A class consists of 12 boys and 15 girls. How many different committees of four can be selected from the class if each committee is to consist of two boys and two girls?

Practice Exercise1. How many combinations of 5 records

can be chosen from 12 records offered by a record club?

2. How many choices of 5 pocketbooks to read can be made from a set of nine pocketbooks?

3. A math professor gave his class a problem set consisting of 10 problems and required each student to answer any 7 problems. In how many ways can a student choose 7 problems from the problem set?