Permutation & COmbination Warm up: What is the probability of getting an ace and a queen assuming replacement does occur? What is the probability of getting a club and a spade assuming replacement does not occur? Suppose 6 people want to sit in 6 chairs in a row at a movie theater. In how many different ways can these 6 people be seated in the 6 available chairs?
Check Homework: • Given Jack, Ace, King, and Queen pick three: • List all possible outcomes (order matter) • Now if order doesn’t matter how many possible outcomes are there? DO YOU SEE A PATTERN IN a) and b) from yesterday and today?
Idea of the day • How do we differentiate when to use permutation or combination rules?
Investigation:How many choices do I have? • Given a King, a Jack, a Queen, and an Ace Task 1: • Students are to pick two cards at a time order does mater. How many possible ways can a student pick 2 cards? Write down your sample space • Assume that order does not matter this time look at your sample space that you’ve just collected. How many possible ways can the student pick two of the same card
Investigation:How many choices do I have? • Given a King, a Jack, a Queen, and an Ace Task 2: • Now let’s pick three cards from the four that are given, assuming order does matter. How many possible ways can a student pick three cards? Write down your sample space • Assuming order doesn’t matter this time. How many ways can a student pick three cards?
Group Discussion: • What have you notice about your sample space on the matter of whether or not order matter? • What conclusion can you make?
Ambassador Game: • Put your name down on a sheet of paper and alphabetize it • The person who are first and last in the list will go to a different group • Then discuss what you’ve discover to see if you all are in agreement with your conclusion
Review: Different Methods • There are 3 methods for calculating the number of possible outcomes for a sequence of event. • Counting Rules • Permutation Rules • Combination Rule
Review: Counting Principle: • Suppose that two events occur in order. If the first can occur in “m” ways and the second in “n” ways (after the first has occurred), then the two events can occur in order in m x n ways. • Order is extremely important for the counting principle. • In simple words, multiply the number of possibilities for each individual event
Permutations • A permutation of a set of distinct objects is an ordering of these objects. • Used for an arrangement of object that are in specific order • The number of permutations of n objects is n!
Example For example, here are some permutations of the word “THEIS”: • SIEHT • HEIST • HTSEI • TISEH etc.
Example: A club has nine members. In how many ways can a president, vice president, and secretary be chosen from the members of this club?
Permutations of n objects taken r at a time The number of permutations of n objects taken r at a time is: On your calculator, permutations are written as nPr and are found under MATHPRB.
Examples: • How many different ways can a chairperson and an assistant chairperson be selected for a research project if there are seven scientists available?
Examples: • How many permutations can be formed from the word “Justice” using only 5 letters?
Example: • From a group of 9 different books, 4 books are to be selected and arranged on a shelf. How many arrangements are possible?
Combinations • Used for an arrangement of objects with No specific order • Order doesn’t matter
Combinations • The formula for computing a combination of nobjects taking rat a time is: • On your calculator, you can use nCr which is located under MATHPRB
Example: • How many combinations of the letter A,B,E,H,L and P are there if they put in groups of 2s?
Example: • At the hamburger Hut you can order hamburgers with cheese, onions, pickles, relish, mustard, lettuce or tomato. How many different combinations of the “extras” can you order, choosing any 3 of them?
Example: • In a club there were 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. Hoe many different possibilities are there?
Revisiting the EQ: • When do I use the counting principle?