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Permutations: Understanding Order and Possibilities

Learn about permutations, factorial notation, and how to calculate the number of ways objects can be arranged. Explore examples and solve problems to improve your understanding.

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Permutations: Understanding Order and Possibilities

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  1. Transparency 3 Click the mouse button or press the Space Bar to display the answers.

  2. Splash Screen

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

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

  5. Example 3-4b Vocabulary Factorial The expression n! is the product of all counting numbers beginning with n and counting backward to 14! = 4 · 3 · 2 · 1

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

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

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

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

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

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

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

  13. 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? The player chosen for first cannot play at second or third P(a, b) = P(10, Permutation = order important Write permutation formula “a” represents the number of choices Replace a with 10 3/3

  14. 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? “b” represents the number wants to choose P(a, b) = P(10, 3) = Replace b with 3 P(10, 3) = 10  Since a = 10, begin the permutation with 10 A permutation is a modified factorial which means to multiply 3/3

  15. 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? b is the number of players want to choose, so multiply 3 numbers counting down from 10 10  9  8 P(a, b) = P(10, 3) = P(10, 3) = 10  9  8 Answer: Multiply P(10, 3) = 720 ways Add dimensional analysis 3/3

  16. Example 3-1b STUDENT COUNCILThere 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

  17. End of Lesson 3 Assignment

  18. Example 3-4a MULTIPLE-CHOICE TEST ITEMConsider 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. probability an even number A 20% B30%C 40% D50% Possible numbers even P (even) = Write probability statement Numerator is in probability statement 4/4

  19. Example 3-4a MULTIPLE-CHOICE TEST ITEMConsider 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. probability an even number A 20% B30%C 40% D50% Denominator is “total numbers possible” Possible numbers even P (even) = Total possible numbers 2 & 4 are the only even numbers 2 P (even) = Replace numerator with 2 4/4

  20. Example 3-4a MULTIPLE-CHOICE TEST ITEMConsider 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(5, 5) = 5  4  3  2  1 5 digits taken 5 at a time P(5, 5) = 120 P(5, 5) = 5! Possible numbers even P (even) = 120 4/4

  21. Example 3-4a MULTIPLE-CHOICE TEST ITEMConsider 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 In order for a number to be even, the ones digit must be 2 or 4. P(4, 4) = 4  3  2  1 P(4, 4) = 24 To write first 4 numbers of an even 5 digit number use permutation 4/4

  22. Example 3-4a MULTIPLE-CHOICE TEST ITEMConsider 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. An even number has to be in the one’s digit P(4, 4) = 24 P(2, 1) = 2 Two digits are even so 2 digits taken 1 at a time P(even) = 24  2 Now multiply the two permutations together P(even) = 48 4/4

  23. Example 3-4a Substitute. 48 P (even) = P(5, 5) = 120 120 P(even) = 48 A 20% B30%C 40% D50% Choices are in % so change probability fraction to a % by dividing numerator by denominator then multiply by 100 P (even) = 40% Answer: C 4/4

  24. Example 3-4b * MULTIPLE-CHOICE TEST ITEMConsider 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% B40%C 50% D60% Answer: D 4/4

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