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

0
875
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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
+22198
0

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}

Mar 20, 2018