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Example 3-4b Objective Find the number of permutations of objects

Example 3-4b Vocabulary Permutation An arrangement or listing in which order is important

Example 3-4b Vocabulary Factorial The expression n! is the product of all counting numbers beginning with n and counting backward to 1 4! = 4 · 3 · 2 · 1

Example 3-4b Math Symbols P(a, b) The number of permutations of a things taken b at a time

Example 3-4b Math Symbols Factorial ! 5! Five factorial 5  4  3  2  1

Lesson 3 Contents Example 1Use Permutation Notation Example 2Use Permutation Notation Example 3Find a Permutation

Example 3-2a Answer: Write permutation Find the value of P(7, 2) = P(7, 2) = 7 Multiply P(7, 2) = 42 1/3 Since 1 st number is the starting number for the multiplication The 2 nd number determines how many numbers to multiply  6

Example 3-2b Answer: P(8, 4) = 1,680 Find the value of 1/3

Example 3-3a Answer: Find the value of P(13, 7) =  12  11  10  9  8  7 P(13, 7) = 8,648,640 2/3 Write permutation Since 1 st number is the starting number for the multiplication P(13, 7) = 13 The 2 nd number determines how many numbers to multiply Multiply

Example 3-3b Answer: P(12, 5) = 95,040 Find the value of 2/3

Example 3-1a SOFTBALL There are 10 players on a softball team. In how many ways can the manager choose three players for first, second, and third base? 3/3 Permutation = order important The player chosen for first cannot play at second or third Write permutation formula P(a, b) = “a” represents the number of choices Replace a with 10 P(10,

Example 3-1a SOFTBALL There are 10 players on a softball team. In how many ways can the manager choose three players for first, second, and third base? 3/3 P(a, b) = P(10, “b” represents the number wants to choose Replace b with 3 3) = Since a = 10, begin the permutation with 10 P(10, 3) = 10 A permutation is a modified factorial which means to multiply 

Example 3-1a SOFTBALL There are 10 players on a softball team. In how many ways can the manager choose three players for first, second, and third base? 3/3 P(a, b) = P(10, b is the number of players want to choose, so multiply 3 numbers counting down from  9  8 3) = P(10, 3) = 10  9  8 Multiply P(10, 3) = 720 Add dimensional analysis ways Answer:

Example 3-1b STUDENT COUNCIL There are 15 students on student council. In how many ways can Mrs. Sommers choose three students for president, vice president, and secretary? Answer: P(15, 3) = 2,730 ways 3/3

End of Lesson 3 Assignment Lesson 8:3Permutations All

Example 3-4a MULTIPLE-CHOICE TEST ITEM Consider all of the five-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that one of these numbers picked at random is an even number. A 20% B 30% C 40% D 50% P (even) = Possible numbers even 4/4 Write probability statement probability an even number Numerator is in probability statement

Example 3-4a MULTIPLE-CHOICE TEST ITEM Consider all of the five-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that one of these numbers picked at random is an even number. A 20% B 30% C 40% D 50% P (even) = Possible numbers even Total possible numbers 4/4 probability an even number Denominator is “total numbers possible” 2 & 4 are the only even numbers Replace numerator with 2 P (even) = 2

Example 3-4a MULTIPLE-CHOICE TEST ITEM Consider all of the five-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that one of these numbers picked at random is an even number. Write permutation P(A, B) = Possible outcomes 5 digits taken 5 at a time P(5, 5) = 5  4  3  2  1 P(5, 5) = 5! P (even) = Possible numbers even 120 P(5, 5) = 120 4/4

Example 3-4a MULTIPLE-CHOICE TEST ITEM Consider all of the five-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that one of these numbers picked at random is an even number. Write permutation P(A, B) = Possible outcomes P(4, 4) = 4  3  2  1 In order for a number to be even, the ones digit must be 2 or 4. To write first 4 numbers of an even 5 digit number use permutation P(4, 4) = 24 4/4

Example 3-4a MULTIPLE-CHOICE TEST ITEM Consider all of the five-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that one of these numbers picked at random is an even number. P(4, 4) = 24 An even number has to be in the one’s digit Two digits are even so 2 digits taken 1 at a time P(2, 1) = 2 P(even) = 24  2 Now multiply the two permutations together P(even) = 48 4/4

Example 3-4a Substitute. P(5, 5) = 120P (even) = P(even) = 48 Choices are in % so change probability fraction to a % by dividing numerator by denominator then multiply by 100 A 20% B 30% C 40% D 50% P (even) = 40% Answer: C 4/4

Example 3-4b MULTIPLE-CHOICE TEST ITEM Consider all of the five-digit numbers that can be formed using the digits 1, 2, 3, 4, and 5 where no digit is used twice. Find the probability that one of these numbers picked at random is an odd number. A 30% B 40% C 50% D 60% Answer: D * 4/4