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Counting

Counting. Just How Many Are There?. Permutations and Combinations. 12.3. Calculate the number of permutations of n objects taken r at a time. Use factorial notation to represent the number of permutations of a set of objects. Permutations and Combinations. 12.3.

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Counting

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  1. Counting Just How Many Are There?

  2. Permutations and Combinations 12.3 • Calculate the number of permutations of n objects taken r at a time. • Use factorial notation to represent the number of permutations of a set of objects.

  3. Permutations and Combinations 12.3 • Calculate the number of combinations of n objects taken r at a time. • Apply the theory of permutations and combinations to solve counting problems.

  4. Permutations

  5. Permutations • Example: How many permutations are there of the letters a, b, c, and d? Write the answer using P(n, r) notation. • Solution: We write the letters a,b,c, and d in a line without repetition, so abcdand bcad are two such permutations. (continued on next slide)

  6. Permutations The slot diagram indicates there are 4 × 3 × 2 × 1 = 24 possibilities. This can be written more succinctly as

  7. Permutations • Example: How many permutations are there of the letters a, b, c, d, e,f,and g if we take the letters three at a time? Write the answer using P(n, r) notation. • Solution: The slot diagram indicates there are 7 × 6 × 5 = 210 possibilities. This can be written in permutation notation as shown.

  8. Factorial Notation

  9. Factorial Notation • Example: Compute (8 – 3)!. • Solution: We work inside parentheses first.

  10. Factorial Notation • Example: Compute . • Solution:

  11. Factorial Notation

  12. Factorial Notation • Example: The 12-person theater group wishes to select one person to direct a play, a second to supervise the music, and a third to handle publicity, tickets, and other administrative details. In how many ways can the group fill these positions? • Solution: This is a permutation of selecting 3 people from 12.

  13. Combinations

  14. Combinations • Example: How many three-element sets can be chosen from a set of five objects? • Solution: Order is not important, so it is clear that this is a combination problem.

  15. Combinations • Example: How many four-person committees can be formed from a set of 10 people? • Solution: Order is not important, so it is clear that this is a combination problem.

  16. Combinations • Example: A syndicate has $15 million to spend on tickets for a lottery. Tickets cost $1 and contain a combination of six numbers from 1 to 44. Does the syndicate have enough money to buy enough tickets to be guaranteed a winner? • Solution: The combination of 6 numbers from the 44 possible gives the number of different tickets. That is, C(44, 6) = 7,059,052. Thus, the syndicate has more than enough money to buy enough tickets to be guaranteed a winner.

  17. Combinations • Example: In the game of poker, five cards are drawn from a standard 52-card deck. How many • different poker hands are possible? • Solution:

  18. Combinations • Example: In the game of bridge, a hand consists of 13 cards drawn from a standard 52-card deck. How many different bridge hands are there? • Solution:

  19. Combining Counting Methods • Example: Two men and two women from a firm will attend a conference. If the firm has ten men and nine women, in how many different ways can the conference attendees be selected? • Solution: The answer is not C(19, 4) since this includes options like four men and no woman being sent to the conference. (continued on next slide)

  20. Combining Counting Methods Stage 1: Select the two women from the nine available. Stage 2: Select the two men from the ten available. Thus, choosing the women and then choosing the men can be done in ways.

  21. Combining Counting Methods • Example: A 16-member consortium wishes to choose a committee consisting of a president, a vice president, and a three-member executive board. In how many different ways can this committee be formed? • Solution: We will count this in two stages: (a) choosing the president and vice president from the consortium, (b) choosing an executive board from the remaining members. (continued on next slide)

  22. Combining Counting Methods Stage 1: Choose the president and vice president. This can be done in P(16, 2) ways. Stage 2: Select the executive board. This can be done in C(14, 3) ways. Total:

  23. Combining Counting Methods

  24. Combining Counting Methods For example, consider the set {1, 2, 3, 4} and the 4th row of Pascal’s triangle: 1 4 6 4 1.

  25. Combining Counting Methods For example, consider the 4th row of Pascal’s triangle: 1 4 6 4 1. C(4, 0) = 1 C(4, 1) = 4 C(4, 2) = 6 C(4, 3) = 4 C(4, 4) = 1

  26. Combining Counting Methods • Example: Assume that a pharmaceutical company has developed five antibiotics and four immune system stimulators. In how many ways can we choose a treatment program consisting of three antibiotics and two immune system stimulators to treat a disease? Use Pascal’s triangle to speed your computations. • Solution: We will count this in two stages: (a) choosing the antibiotics, (b) choosing the immune system simulators. (continued on next slide)

  27. Combining Counting Methods Stage 1: Choosing 3 antibiotics from 5 can be done in C(5, 3) ways. Stage 2: Choosing 2 immune system simulators from 4 can be done in C(4, 2) ways. Total: C(5, 3) × C(4, 2) = 10 × 6 = 60 ways.

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