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How many subsets of the set $\{1,2,3,4,\ldots,10\}$ have no two consecutive numbers as members?

 

Its not 50

 Mar 20, 2018
 #1
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How many subsets of the set $\{1,2,3,4,\ldots,10\}$ have no two consecutive numbers as members?

 

\(\begin{array}{|l|cr|r|} \hline & &&\text{okay} \\ \hline \binom{10}{0}\times \ 0-\text{elements-subsets } &=& 1 & 1 \\ \binom{10}{1}\times \ 1-\text{elements-subsets } &=& 10 & 10 \\ \binom{10}{2}\times \ 2-\text{elements-subsets } &=& 45 & 36 \\ \binom{10}{3}\times \ 3-\text{elements-subsets } &=& 120 & 56 \\ \binom{10}{4}\times \ 4-\text{elements-subsets } &=& 210 & 35 \\ \binom{10}{5}\times \ 5-\text{elements-subsets } &=& 252 & 6 \\ \binom{10}{6}\times \ 6-\text{elements-subsets } &=& 210 & 0 \\ \binom{10}{7}\times \ 7-\text{elements-subsets } &=& 120 & 0 \\ \binom{10}{8}\times \ 8-\text{elements-subsets } &=& 45 & 0 \\ \binom{10}{9}\times \ 9-\text{elements-subsets } &=& 10 & 0 \\ \binom{10}{10}\times \ 10-\text{elements-subsets } &=& 1 & 0 \\ \hline \text{sum } &=& & 144 \\ \hline \end{array}\)

 

The subsets:

   1. { }

   2. {1}
   3. {2}
   4. {3}
   5. {4}
   6. {5}
   7. {6}
   8. {7}
   9. {8}
  10. {9}
  11. {10}

  12. {1, 3}
  13. {1, 4}
  14. {1, 5}
  15. {1, 6}
  16. {1, 7}
  17. {1, 8}
  18. {1, 9}
  19. {1, 10}
  20. {2, 4}
  21. {2, 5}
  22. {2, 6}
  23. {2, 7}
  24. {2, 8}
  25. {2, 9}
  26. {2, 10}
  27. {3, 5}
  28. {3, 6}
  29. {3, 7}
  30. {3, 8}
  31. {3, 9}
  32. {3, 10}
  33. {4, 6}
  34. {4, 7}
  35. {4, 8}
  36. {4, 9}
  37. {4, 10}
  38. {5, 7}
  39. {5, 8}
  40. {5, 9}
  41. {5, 10}
  42. {6, 8}
  43. {6, 9}
  44. {6, 10}
  45. {7, 9}
  46. {7, 10}
  47. {8, 10}

  48. {1, 3, 5}
  49. {1, 3, 6}
  50. {1, 3, 7}
  51. {1, 3, 8}
  52. {1, 3, 9}
  53. {1, 3, 10}
  54. {1, 4, 6}
  55. {1, 4, 7}
  56. {1, 4, 8}
  57. {1, 4, 9}
  58. {1, 4, 10}
  59. {1, 5, 7}
  60. {1, 5, 8}
  61. {1, 5, 9}
  62. {1, 5, 10}
  63. {1, 6, 8}
  64. {1, 6, 9}
  65. {1, 6, 10}
  66. {1, 7, 9}
  67. {1, 7, 10}
  68. {1, 8, 10}
  69. {2, 4, 6}
  70. {2, 4, 7}
  71. {2, 4, 8}
  72. {2, 4, 9}
  73. {2, 4, 10}
  74. {2, 5, 7}
  75. {2, 5, 8}
  76. {2, 5, 9}
  77. {2, 5, 10}
  78. {2, 6, 8}
  79. {2, 6, 9}
  80. {2, 6, 10}
  81. {2, 7, 9}
  82. {2, 7, 10}
  83. {2, 8, 10}
  84. {3, 5, 7}
  85. {3, 5, 8}
  86. {3, 5, 9}
  87. {3, 5, 10}
  88. {3, 6, 8}
  89. {3, 6, 9}
  90. {3, 6, 10}
  91. {3, 7, 9}
  92. {3, 7, 10}
  93. {3, 8, 10}
  94. {4, 6, 8}
  95. {4, 6, 9}
  96. {4, 6, 10}
  97. {4, 7, 9}
  98. {4, 7, 10}
  99. {4, 8, 10}
 100. {5, 7, 9}
 101. {5, 7, 10}
 102. {5, 8, 10}
 103. {6, 8, 10}

 104. {1, 3, 5, 7}
 105. {1, 3, 5, 8}
 106. {1, 3, 5, 9}
 107. {1, 3, 5, 10}
 108. {1, 3, 6, 8}
 109. {1, 3, 6, 9}
 110. {1, 3, 6, 10}
 111. {1, 3, 7, 9}
 112. {1, 3, 7, 10}
 113. {1, 3, 8, 10}
 114. {1, 4, 6, 8}
 115. {1, 4, 6, 9}
 116. {1, 4, 6, 10}
 117. {1, 4, 7, 9}
 118. {1, 4, 7, 10}
 119. {1, 4, 8, 10}
 120. {1, 5, 7, 9}
 121. {1, 5, 7, 10}
 122. {1, 5, 8, 10}
 123. {1, 6, 8, 10}
 124. {2, 4, 6, 8}
 125. {2, 4, 6, 9}
 126. {2, 4, 6, 10}
 127. {2, 4, 7, 9}
 128. {2, 4, 7, 10}
 129. {2, 4, 8, 10}
 130. {2, 5, 7, 9}
 131. {2, 5, 7, 10}
 132. {2, 5, 8, 10}
 133. {2, 6, 8, 10}
 134. {3, 5, 7, 9}
 135. {3, 5, 7, 10}
 136. {3, 5, 8, 10}
 137. {3, 6, 8, 10}
 138. {4, 6, 8, 10}

 139. {1, 3, 5, 7, 9}
 140. {1, 3, 5, 7, 10}
 141. {1, 3, 5, 8, 10}
 142. {1, 3, 6, 8, 10}
 143. {1, 4, 6, 8, 10}
 144. {2, 4, 6, 8, 10}

 

laugh

 Mar 20, 2018

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