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Counting Techniques 1

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Sequential Counting Principle Section 10.1 2

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Tree Diagrams Tree Diagrams are used to display all the possible outcomes of a situation. T H T H H T 3

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Sequential Counting Principle SCP If an event can occur in m ways and a second event can occur in n ways, then the sequence of two events can occur in m x n ways. 4

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SCP Fundamental Counting Principle If an event can occur in m ways, and then a second event can occur in n ways, and then a third event can occur in r ways, and so on, then the sequence of events can occur in m n n r … ways. 5

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Exercise: A bag contains 6 brown marbles and 3 white marbles. You are asked to draw 3 marbles from the bag without replacement. In how many ways can you draw 2 brown and 1 white in that order? 6 x 5 x 3 = 90 1 st B2 nd B 1 st W 6

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Exercise: In how many ways can 1 woman and 1 man, in that order, be selected from 5 women and 6 men? 5 x 6 = 30 7

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Exercise: How many different sets of 3 initials can be made from the letters of the English alphabet? Hint:AAA is different from BBB. 26 x x = 17,576 8

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Exercise: The Good Taste Restaurant has 7 entrees, 6 vegetables, and 9 desserts on its menu. If you want to order 1 entree, 1 vegetable, and 1 dessert, how many choices do you have? 7 x 6 x 9 = 378 9

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In Connecticut, auto license plates carry 3 digits followed by 3 letters. a. How many arrangements are possible for the 3 letters? b. How many arrangements are possible for the 3 numbers? c. How many different license plates can be made using 3 numbers followed by 3 letters? Exercise: a. 26 x x = 17,576 b. 10 x x = 1000 c. 17,576 x 1000 = 17,576,000 10

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Social Security numbers consist of nine digits. a. If the first digit cannot be a 0, how many Social Security numbers are possible? b. How many Social Security numbers are possible if there are no restrictions? Exercise: 9 x 10 8 = 900,000,000 10 9 = 1,000,000,000 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11

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You are expected to be familiar with a standard deck of playing cards. Standard deck 52 cards Black 26Red 26 13 Spades ♥ 13 Hearts ♣ 13 Clubs ♦ 13 Diamonds 12 Face Cards (Jack, Queen, & King) Black 6Red 6 3 Spades ♥ 3 Hearts ♣ 3 Clubs ♦ 3 Diamonds 40 Non - Face Cards (A -10) Black 20Red 20 10 Spades ♥ 10 Hearts ♣ 10 Clubs ♦ 10 Diamonds 13

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Note: the Ace will be considered to be a one. There are four suits: Clubs, Diamonds, Spades, and Hearts in a deck of cards. END 14

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Draw 3 cards without replacement from a standard 52 card deck. What is the probability that: 1.They are all red ? 2.At least one is black ? 3.They are.

Draw 3 cards without replacement from a standard 52 card deck. What is the probability that: 1.They are all red ? 2.At least one is black ? 3.They are.

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