Study Notes Finite Mathematics M118: Combinations Permutations, Study notes of Mathematics

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Try this warm-up problem:

You flip a coin 3 times. Construct the set S which contains all the possible sequences of heads and tails on the 3 flips.

Chapter 3: Combinatorics

Chapter 3 is a challenging chapter and critical for Chapter 4, another challenging chapter for many students. Combinatorics is about counting. Our focus: finding the number of different outcomes an experiment can have (counting)

Sample Space

• (^) The sample space for an experiment is the set of all possible outcomes. We saw this in the warm-up when we listed all possible outcomes from flipping 3 coins.

3.1 Trees and Equally Likely Outcomes

Counting Techniques

• (^) For a simple experiment, we can list all possible outcomes.
• (^) For more complicated experiments, the fundamental counting principle or a tree diagram may help.
• (^) When there are too many possible outcomes to list or diagram, we will use sophisticated counting techniques: permutations (section 3.2) and combinations (3.3).

Tree Diagrams

• (^) Example: Kids wear uniforms to school. Pants must be navy or khaki. Polo shirts must be white, red or light blue. How many different outfits are possible?
• (^) Use the fundamental counting principle.
• (^) Now use a tree diagram.
• (^) Now add in a fashion restriction.

How many 4 digit numbers can be formed

using the digits 1,2,3,5,7,8,9 if:

a) Digits cannot be repeated? b) Repetition of digits IS allowed? c) Repetition is allowed and the 4-digit number must be even? d) Repetition of digits is NOT allowed and the 4-digit number must be even? e) No repetition and the 4-digit number must be greater than 5000?

Factorial

n! = 1! = 0! = Using a calculator.

You have 3 Skittles in your pocket, 1 red and 2 green. You reach into your pocket and randomly select and eat Skittles until you have eaten the red one.

• (^) Draw a tree diagram to illustrate the possible outcomes of this experiment.
• (^) How many outcomes are in the sample space?
• (^) List the sample space for this experiment. Try this one again, this time with 4 Skittles, 1 red, 2 green and 1 yellow.

3.2 Permutations and

3.3 Combinations

• (^) Permutations are ordered outcomes (order matters): Select a president, VP and secretary.
• (^) Combinations are unordered outcomes (order does not matter). Select a committee of 3.

In how many ways can a group of 20

select a president, VP and secretary?

• (^) Order matters
• (^) Use the fundamental counting principle
• (^) This is called a permutation of 20 objects taken 3 at a time.
• (^) P(20,3) = 20 * 19 * 18 = 6,
• (^) On my calculator: 20 nPr 3 =

There are 10 people who apply for 4 job openings at the new dining hall that will open next year in the old University Place hotel. In how many ways can the 4 openings be filled if: a.The jobs are identical? b.The jobs are all different? Let’s look at the general formulas for permutations and combinations and compare them.

From my class of 25 students,

in how many ways can I :

a) Choose a hall monitor and an assistant hall monitor? b) Choose 2 students to help clean the classroom? c) Choose 3 students to win 3 identical prizes if a student cannot be a repeat winner? d) Choose 3 students to win 3 different prizes if a student cannot be a repeat winner? e) Choose 3 students to win 3 different prizes if a student IS ALLOWED to be a repeat winner?

An urn contains 4 red and 2 blue marbles. We choose 2 marbles without replacement. In how many ways can we choose: Two red marbles? One red and one blue marble? No red marbles Any two marbles? We call this the case with “no restrictions.” At least one red marble? (It might help to number or name the marbles and to use a table to keep track of possibilities)

There are 3 Independents, 4 Democrats, and 5 Republicans: In how many ways a. Can the 12 people be seated in a row of chairs if there are no restictions? b. Can the 12 people be seated in a row of chairs if the Democrats must sit together on the left, the Republicans must sit together on the right, and the Independents must sit together in the middle? c. Can we choose a committee of 3 if the committee must contain exactly 1 Democrat? d. Can we choose a committee of 3 if there are no restrictions? e. Can we choose 4 different people to win 4 different prizes if the biggest prize must go to an Independent? f. Choose 4 different people to win 4 identical prizes? g. Choose 3 co-presidents, if they must come from different parties? h. Choose 3 co-presidents, if they must come from the same party?