Counting Principles

Counting Principles The Fundamental Counting Principle: If one event can occur m ways and another can occur n ways, then the number of ways the events can occur in sequence is m*n. ******************************************** Example: A die roll can result in six different outcomes: 1,2,3,4,5,6. A coin flip can result in 2 different outcomes: H or T A die roll and a coin flip can result in 2*6 = 12 different outcomes: 1H, 2H, 3H, 4H, 5H, 6H, 1T, 2T, 3T, 4T, 5T, 6T Pictures: http://commons.wikimedia.org/

Counting Principles Example: License Plates have 3 digits followed by 3 letters. How many license plates are possible? __ __ __ __ __ __

Counting Principles Example: License Plates have 3 digits followed by 3 letters. How many license plates are possible? __ __ __ __ __ __ 10*10*10*26*26*26 = 17,576,000

Counting Principles Example: You have 2 pairs of socks, 3 pairs of pants and 2 shirts. How many different outfits can you make? ___ ___ ___ socks shirts pants

Counting Principles Example: You have 2 pairs of socks, 3 pairs of pants and 2 shirts. How many different outfits can you make? _2_*_3_*_2_ = 12 socks pants shirts

Factorials N! = n*(n-1)*(n-2)…1 Examples: 5! = 5*4*3*2*1 = 120 7! = 7*6*5*4*3*2*1 =5040 Counting Principles

Counting Principles Permutation: An ordered arrangement of objects (no repetition and order matters) Example:

Counting Principles Permutation: An ordered arrangement of objects (no repetition and order matters) Another way to look at it: Three slots and five objects to choose from to fill them without replacement: ___*___*___

Counting Principles Example: Permutation: An ordered arrangement of objects (no repetition and order matters) _5_ _4_ _3_= 60

Counting Principles Example: Permutation: An ordered arrangement of objects (no repetition and order matters) How many ways can five people finish a race 1st, 2nd and 3rd? _5_*_4_*_3_= 60 first second third

Counting Principles Combination: selection of r objects from a group of n objects (no repetition and order does not matter) Notice that this is the same formula as for a permutation, but you divide by r! because order does not matter and the objects can be ordered in r! ways

Counting Principles 5 4 3 Combination: selection of r objects from a group of n objects (no repetition and order does not matter) ________________ = 60/6 = 10 3!

Counting Principles Combination: selection of r objects from a group of n objects (no repetition and order does not matter) How many ways are there to choose a three member team from five people? ________________ = 60/6 = 10 3! 5 4 3 Divide by the number of ways to order three objects

Counting Principles Distinguishable Permutations If there are n1 of one type of object and n2 of another type and there are n total, then there are distinguishable ways of arranging them. Example: How many distinguishable ways can you arrange AAABB?

Counting Principles Distinguishable Permutations Example: How many distinguishable ways can you arrange the letters in Mississippi?

Probability How many ways can you be dealt a five diamond hand from a deck of cards? We choose five cards from the 13 diamonds, then divide by the number of ways to choose five cards from all 52 We can apply these rules to probability:

Probability How many ways can you be dealt any flush from a deck of cards? First choose the suite from 4 suites, then choose five cards from 13 of that suite: We can apply these rules to probability:

Probability How many ways can you be dealt a full house from a deck of cards? First choose the card for the three of a kind. Then, choose 3 of those cards, then choose the card for the two of a kind, then choose two of those cards: We can apply these rules to probability:

Probability Another example: You have 25 students in a class. 20 are passing. You choose 5 students. What is the probability you choose three passing students and two failing? A distribution called the hypergeometric distribution is based on these kind of situations. We won’t worry about that now.

Counting Principles