Fundamental Counting Theory, Permutations and Combinations

Fundamental Counting Theory, Permutations and Combinations DM. 13

The Fundamental Counting Theory A method for counting outcomes of multi-stage processes If you want to perform a series of tasks and the first task can be done in (a) ways, the second can be done in (b) ways, the third can be done in (c) ways, and so on, then all the tasks can be done in a x b x c…ways

Example In a restaurant there are four kinds of soup, twelve entrees, six desserts and three drinks. How many different four course meals can we choose?

Example How many ways can four coins be flipped?

Example How many ways can three dice (red, green, and yellow) be rolled?

Example Matthew is working as a summer intern for a TV station and wants to vary his outfit by wearing different combinations of coats, pants, shirts and ties. If he has three sports coats, five pairs of pants, seven shirts and four ties. How many way can he choose an outfit with a coat, pants, shirt and tie?

Example A California license plate starts with a digit other than 0 followed by three capital letters followed by three more digits (0 through 9). • How many different license plates are possible? • How many different license plates start with 5 and end with 9? • How many have no repeated symbols (all digits are different and all the letters are different)?

Code words Code words: Examples: Discrete Mississippi Letters Number of Letters Repeated Letters

Permutations A group of objects in which the ordering of the objects within the group makes a difference. If we select (r) different objects from a set of (n) objects and arrange them in a straight line, this is called a permutation of n objects The number of permutations of (n) objects taken (r) at a time is denoted p(n,r) Formula: nPr = Application: Stud Poker, Rankings, Committees with assignments

Example How many Permutations are there of the letters a,b,c,d?

Example How many Permutations are there of the letters a, b, c, d, e, f and g if we take the letters three at a time?

Combinations A group of objects in which the order of the objects is irrelevant When choosing (r) objects from a set of (n) objects, we say that we are forming a combination of (n) objects (r) at a time. Noted: c(n, r) Formula: Applications: Draw Poker hands, lottery tickets, Coalitions, subsets.

Example How many 3 –element sets can be chosen from a set of five objects?

Example How many four person committees can be formed from a set of ten people?

How many tickets are necessary to cover all possibilities in a lottery? A group of people intend to raise $15 million to buy all the ticket combinations for certain lotteries. The plan was that if the prize was larger than the amount spent on the tickets, then the group would be guaranteed a profit. To play the Virginia lottery, the player buys a ticket for $1 containing a combination of six numbers from 1 to 44. Assuming they raised the $15 million, does the group have enough money to buy enough tickets to be guaranteed a winner?

Example In the game of poker, 5 cards are drawn from a standard 52 – card deck. How many different poker hands are possible?

Fundamental Counting Theory, Permutations and Combinations