Counting Rules.

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Presentation transcript:

Counting Rules

Fundament Counting Rule In a sequence of 𝑛 events in which the first has 𝑘 1 possibilities and the second event has 𝑘 2 and the third has 𝑘 3 , and so on. The total number of possibilities will be 𝑘 1 ⋅ 𝑘 2 ⋅ 𝑘 3 ⋅⋅⋅ 𝑘 𝑛

Fundament Counting Rule A coin is tossed and a die is rolled. Find the number of outcomes for the sequence of events.

Fundament Counting Rule A coin is tossed and a die is rolled. Find the number of outcomes for the sequence of events. 2⋅6=12

Fundament Counting Rule Assume that a criminal is found using your social security number and claims that all of the digits were randomly generated. What is the probability of getting your social security number when randomly generating nine digits?

Fundament Counting Rule Assume that a criminal is found using your social security number and claims that all of the digits were randomly generated. What is the probability of getting your social security number when randomly generating nine digits? 9 digits each with 10 outcomes is 10 9 =1,000,000,000 So 1 1,000,000,000

Fundament Counting Rule Consider the following problem given on a history test: Arrange the following events in chronological order. Boston Tea Party Teapot Dome Scandal The Civil War The correct answer is a, c, b. What is the probability of guessing the correct order?

Factorials The Factorial symbol (!) denotes the product of decreasing positive whole numbers. Ex: 4!=4⋅3⋅2⋅1=24. By definition 0!=1

Factorial Rule A collection of 𝑛 different items can be arranged in 𝑛! Different ways.

Factorial Rule/Permutations During the summer, you are planning to visit these six national parks: Glacier, Yellowstone, Yosemite, Arches, Zion, and Grand Canyon. You would like to find the most efficient route. How many different routes are possible?

Permutations Now say we only have time to visit 4 of the 50 state capitals what are the number of different routes?

Permutations Now say we only have time to visit 4 of the 50 state capitals what are the number of different routes? 50! 46! =50⋅49⋅48⋅47=5,527,200

Permutations The arrangement of 𝑛 objects in a specific order using r objects at a time is written with formula 𝑛 𝑃 𝑟 = 𝑛! 𝑛−𝑟 !

Permutations In horse racing, a bet on an exacta in a race is won by correctly selecting the horses that finish first and second, and you must select those two horses in the correct order. A horse race has 20 horses, if you randomly select two horses what are you chances of winning a bet on an exacta?

Permutations In horse racing, a bet on an exacta in a race is won by correctly selecting the horses that finish first and second, and you must select those two horses in the correct order. A horse race has 20 horses, if you randomly select two horses what are you chances of winning a bet on an exacta? 20! 20−2 ! = 20! 18! =20⋅19=380, so 1/380

Permutations In horse racing, a bet on an exacta in a race is won by correctly selecting the horses that finish first and second, and you must select those two horses in the correct order. A horse race has 20 horses, if you randomly select two horses what are you chances of winning a bet on an exacta? 20! 20−2 ! = 20! 18! =20⋅19=380, so 1/380

Combinations A selection of distinct objects without regard to order is called a combination.

Combinations A selection of distinct objects without regard to order is called a combination. Given the letters A, B, C, and D, list the permutations and combinations for selecting two letters.

Combinations The number of combinations of 𝑟 objects selected from n objects is denoted by 𝑛 𝐶 𝑟 or 𝑛 𝑟 = 𝑛! 𝑛−𝑟 !𝑟!

Combinations How many combinations of 4 objects taken 2 at a time? 4 2 = 4! 4−2 !2! = 4! 2!2! = 4⋅3 2⋅1 =6

Combinations The Foreign Language Club is showing a four-movie marathon of subtitled movies. How many ways can they choose 4 from the 11 available?

Combinations The Foreign Language Club is showing a four-movie marathon of subtitled movies. How many ways can they choose 4 from the 11 available? 11 4 = 11! 11−4 !4! = 11⋅10⋅9⋅8⋅7⋅6 ⋅5 4⋅3⋅2⋅1 =15

Combinations How many ways can a jury of 6 women and 6 men be selected from 10 women and 12 men?

Combinations How many ways can a jury of 6 women and 6 men be selected from 10 women and 12 men? 10 6 ⋅ 12 6 =16⋅18=224

Homework! 4-4: 1-59 odd