If replacement is allowed, find all:

  1. 5-permutations of a 2-set

Please note that my solution differs from the authors’ one given at the end

  1. \(\{ a, a, a, a, a \}\)
  2. \(\{ a, a, a, a, b \}\)
  3. \(\{ a, a, a, b, a \}\)
  4. \(\{ a, a, a, b, b \}\)
  5. \(\{ a, a, b, a, a \}\)
  6. \(\{ a, a, b, a, b \}\)
  7. \(\{ a, a, b, b, a \}\)
  8. \(\{ a, a, b, b, b \}\)
  9. \(\{ a, b, a, a, a \}\)
  10. \(\{ a, b, a, a, b \}\)
  11. \(\{ a, b, a, b, a \}\)
  12. \(\{ a, b, a, b, b \}\)
  13. \(\{ a, b, b, a, a \}\)
  14. \(\{ a, b, b, a, b \}\)
  15. \(\{ a, b, b, b, a \}\)
  16. \(\{ a, b, b, b, b \}\)
  17. \(\{ b, a, a, a, a \}\)
  18. \(\{ b, a, a, a, b \}\)
  19. \(\{ b, a, a, b, a \}\)
  20. \(\{ b, a, a, b, b \}\)
  21. \(\{ b, a, b, a, a \}\)
  22. \(\{ b, a, b, a, b \}\)
  23. \(\{ b, a, b, b, a \}\)
  24. \(\{ b, a, b, b, b \}\)
  25. \(\{ b, b, a, a, a \}\)
  26. \(\{ b, b, a, a, b \}\)
  27. \(\{ b, b, a, b, a \}\)
  28. \(\{ b, b, a, b, b \}\)
  29. \(\{ b, b, b, a, a \}\)
  30. \(\{ b, b, b, a, b \}\)
  31. \(\{ b, b, b, b, a \}\)
  32. \(\{ b, b, b, b, b \}\)

Authors’ answer: \(aaaa, aaab, aaba, abaa, baaa, aabb, abab, abba, baab, baba, bbaa, abbb, babb, bbab, bbba, bbbb;\)1

  1. 2-permutations of a 3-set
  1. \(\{ a, a \}\)
  2. \(\{ a, b \}\)
  3. \(\{ a, c \}\)
  4. \(\{ b, a \}\)
  5. \(\{ b, b \}\)
  6. \(\{ b, c \}\)
  7. \(\{ c, a \}\)
  8. \(\{ c, b \}\)
  9. \(\{ c, c \}\)
  1. 5-combinations of a 2-set

Please note that my solution differs from the authors’ one given at the end

  1. \(\{ a, a, a, a, a \}\)
  2. \(\{ a, a, a, a, b \}\)
  3. \(\{ a, a, a, b, b \}\)
  4. \(\{ a, a, b, b, b \}\)
  5. \(\{ a, b, b, b, b \}\)
  6. \(\{ b, b, b, b, b \}\)

Authors’ answer: \(\{a, a, a, a\}, \{a, a, a, b\}, \{a, a, b, b\}, \{a, b, b, b\}, \{b, b, b, b\};\)2

  1. 2-combinations of a 3-set
  1. \(\{ a, a \}\)
  2. \(\{ a, b \}\)
  3. \(\{ a, c \}\)
  4. \(\{ b, b \}\)
  5. \(\{ b, c \}\)
  6. \(\{ c, c \}\)

  1. Clearly a typo, the authors posted 4-permutations, instead of 5-permutations, of a 2-set↩

  2. Similarly looks like a typo, the authors posted 4-combinations, instead of 5-combinations, of a 2-set↩