Virtual Laboratories > Combinatorics > 1 2 3 4 [5]

5. Notes

Combinatorics is a rich and beautiful mathematical subject in its own right, not just because of its connection to probability.

Books

Selected Answers for Section 1

Answer 1.9.

  1. 67,600,000
  2. 1 / 32

Answer 1.10.

  1. 1,679,616
  2. 264 / 364 ~ 0.2721

Answer 1.11 

  1. 21 cards
  2. 324 outcomes
  3. The best hand would be the 5 remaining weapons or the 5 remaining suspects.

Answer 1.13.

  1. 624
  2. 1 / 16

Answer 1.15.

  1. 7776
  2. 1 / 36

Answer 1.16.

  1. 41,969,002,243,198,805,166,015,625
  2. 0.1953 × 10-10.

Answer 1.20.

  1. 1024
  2. 1 / 8

Answer 1.21. 1048576

Answer 1.22.

  1. 126
  2. 21 / 128.

Answer 1.31. 1 / 2k.

Selected Answers for Section 2

Answer 2.4. 720

Answer 2.5.

  1. 40,320
  2. 1152
  3. 2880
  4. 384

Answer 2.7.

  1. 479,001,600
  2. 103,680

Answer 2.8. Let pn denote the event that the birthdays of the n people are distinct.

  1. pn = (365)n / 365n.
  2. We assume that birthdays are uniformly distributed throughout the year.
  3. p10 = 0.8831, p20 = 0.5886, p30 = 0.2937, p40 = 0.1088.

Answer 2.10. 189 / 624

Answer 2.13. 5 / 54.

Answer 2.14. 189 / 650.

Answer 2.15.

  1. -210
  2. -15 / 16
  3. -3640 / 243

Selected Answers for Section 3

Answer 3.4.

  1. 3744 / 2598960 = 0.001441.
  2. 624 / 2598860 = 0.000240.

Answer 3.5.

  1. 0.238608.
  2. 0.0741397
  3. 0.017959

Answer 3.7.

  1. 38,760
  2. 13,860
  3. 30,800

Answer 3.8. 1,913,496

Answer 3.9.

  1. 1.41662 × 10-7.

Answer 3.15.

  1. 210 / 1024.
  2. 56 / 1024.

Answer 3.16. 6160 / 15504 = 0.297317.

Answer 3.23. 108,864

Answer 3.24. 71,680

Answer 3.29. 364

Answer 3.32.

  1. 1771
  2. 969

Answer 3.33. 252

Answer 3.34.

  1. 66
  2. 36

Answer 3.35.

  1. With replacement, with order: 10,000
  2. With replacement, without order: 715
  3. Without replacement, with order: 5040
  4. Without replacement, without order: 210

Answer 3.36.

  1. 1 / 16.
  2. 70
  3. -91 / 729.

Selected Answers for Section 4

Answer 4.5. 9,777,287,520

Answer 4.8.

  1. 50,400
  2. 9,979,200
  3. 34,650
  4. 3780
  5. 210

Answer 4.9. 27,720

Answer 4.12. 360,360

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