Counting

Counting The idea is to find out how many members are present in a finite set. Principles of counting answer the following kind of questions: How many four digit numbers can there be if repetition of numbers are allowed and if repetition of numbers are not allowed? If a man has 4 suits, 8 shirts and 5 ties, how many outfits can he put together?

Multiplication Principle Principle: If there are n possible outcomes for a first event and m possible outcomes for a second event, then there are n x m possible outcomes for the sequence of two events. Hence, from the multiplication principle, it follows that for two sets A and B |AB| = |A| x |B|

Example A child is allowed to choose one jellybean out of two jellybeans, one red and one black, and one gummy bear out of three gummy bears, yellow, green, and white. How many different sets of candy can the child have?

Solving the Example There are 23=6 or 32=6 possible outcomes as seen from the following figures

Exercises Exercise 1 Exercise 2 The last part of your telephone number contains four digits. How many such four-digit numbers are there? What if the same digit cannot be used twice? Exercise 2 A video game is begun by making selections from each of three menus. The first menu (number of players) has four selections, the second menu (level of play) has eight, and the third menu (speed) has six. In how many configurations can the game be played?

Addition Principle Addition Principle: If A and B are disjoint events with n and m outcomes, respectively, then the total number of possible outcomes for event “A or B” is n+m. If A and B are disjoint sets, then |A  B| = |A| + |B| using the addition principle. Example A customer wants to purchase a vehicle from a dealer. The dealer has 23 autos and 14 trucks in stock. How many selections does the customer have?

Exercises Exercise 1 Exercise 2 How many four-digit numbers begin with a 4 or a 5? Exercise 2 A president and vice-president must be chosen for the executive committee of an organization. There are 17 volunteers from the Eastern Division and 24 volunteers from the Western Division. If both officers must come from the same division, in how many ways can the officers be selected?

Decision trees Trees that provide the number of outcomes of an event based on a series of possible choices are called decision trees. Tony is pitching pennies. Each toss results in heads (H) or tails (T). How many ways can he toss the coin five times without having two heads in a row? There are 13 possible outcomes as seen from the tree.

Exercises Exercise 1 Exercise 2 Draw a decision tree to find the number of binary strings of length 4 that do not have consecutive 0s. Exercise 2 Voting on a certain issue is conducted by having everyone put a red, blue or green slip of paper into a hat. Then the slips are pulled out one at a time. The first color to receive two votes wins. Draw a decision tree to find the number of ways in which the balloting can occur.