Finding Sets - Difficult one !

[sandip.dhamapurkar]

Combination of TWO
===============

I have four basic numbers 1 2 3 4. If I have to make all possible
combinations of TWO out of these four basic numbers, then the answer
would be SIX =COMBIN(4,2).

I am planning to make unique sets out of these combinations. Since
there are four basic numbers, I will have to make a grid of 2 X 2
consisting all unique numbers which comes out to be THREE

1. 1-2 & 3-4 (All individual numbers in each set is unique)
2. 1-3 & 2-4
3. 1-4 & 2-3

If I have six basic numbers 1 2 3 4 5 6 and if I have to make all
possible combinations of TWO, then the answer would be FIFTEEN
=COMBIN(6,2)

If I make unique sets out of these combinations, I will have to make a
grid of 3 X 2 which comes out to be FIFTEEN

1. 1-2 & 3-4 & 5-6 (All individual numbers in each set is unique)
2. 1-2 & 3-5 & 4-6
3. 1-2 & 3-6 & 4-5
4. 1-3 & 2-4 & 5-6
5. 1-3 & 2-5 & 4-6
6. 1-3 & 2-6 & 4-5
7. 1-4 & 2-3 & 5-6
8. 1-4 & 2-5 & 3-6
9. 1-4 & 2-6 & 3-5
10. 1-5 & 2-3 & 4-6
11. 1-5 & 2-4 & 3-6
12. 1-5 & 2-6 & 3-4
13. 1-6 & 2-3 & 4-5
14. 1-6 & 2-4 & 3-5
15. 1-6 & 2-5 & 3-4

Combination of THREE
=================

I have six basic numbers 1 2 3 4 5 6. If I have to make all possible
combinations of THREE out of these six basic numbers, then the answer
would be TWENTY =COMBIN(3,3).

If I have to make unique sets out of these combinations. Since there
are six basic numbers, I will have to make a grid of 3 X 2 consisting
all unique numbers which comes out to be TEN

1. 1-2-3 & 4-5-6 (All individual numbers in each set is unique)
2. 1-2-4 & 3-5-6
3. 1-2-5 & 3-4-6
4. 1-2-6 & 3-4-5
5. 1-3-4 & 2-5-6
6. 1-3-5 & 2-4-6
7. 1-3-6 & 2-4-5
8. 1-4-5 & 2-3-6
9. 1-4-6 & 2-3-5
10. 1-5-6 & 2-3-4

If I have Nine basic numbers 1 2 3 4 5 6 7 8 9 and if I have to make
all possible combinations of THREE, then the answer would be EIGHTY
FOUR =COMBIN(9,3)

If I make unique sets out of these combinations, I will have to make a
grid of 3 X 3 which comes out to be 280

1. 1-2-3 & 4-5-6 & 7-8-9 (All individual numbers in each set is unique)
2. 1-2-3 & 4-5-7 & 6-8-9
..
..
..
..
..
..
280. 1-8-9 & 2-6-7 & 3-4-5

X------------X------ End of the problem ------X------------X

Answer for first scenario was 3
Answer for second scenario was 15
Answer for third scenario was 10
Answer for last scenario was 280

What I want to know is whether there is any mathematical/statistical
formula to calculate the result. I want to know is there any way to
derive the answer from any formula?

Sandy

 

[sandip.dhamapurkar]

Had a hard time listing these 280 sets manually. Anyways, here are
they:

01. 1-2-3 & 4-5-6 & 7-8-9 | 141. 1-4-6 & 2-3-5 & 7-8-9
02. 1-2-3 & 4-5-7 & 6-8-9 | 142. 1-4-6 & 2-3-7 & 5-8-9
03. 1-2-3 & 4-5-8 & 6-7-9 | 143. 1-4-6 & 2-3-8 & 5-7-9
04. 1-2-3 & 4-5-9 & 6-7-8 | 144. 1-4-6 & 2-3-9 & 5-7-8
05. 1-2-3 & 4-6-7 & 5-8-9 | 145. 1-4-6 & 2-5-7 & 3-8-9
06. 1-2-3 & 4-6-8 & 5-7-9 | 146. 1-4-6 & 2-5-8 & 3-7-9
07. 1-2-3 & 4-6-9 & 5-7-8 | 147. 1-4-6 & 2-5-9 & 3-7-8
08. 1-2-3 & 4-7-8 & 5-6-9 | 148. 1-4-6 & 2-7-8 & 3-5-9
09. 1-2-3 & 4-7-9 & 5-6-8 | 149. 1-4-6 & 2-7-9 & 3-5-8
10. 1-2-3 & 4-8-9 & 5-6-7 | 150. 1-4-6 & 2-8-9 & 3-5-7
11. 1-2-4 & 3-5-6 & 7-8-9 | 151. 1-4-7 & 2-3-5 & 6-8-9
12. 1-2-4 & 3-5-7 & 6-8-9 | 152. 1-4-7 & 2-3-6 & 5-8-9
13. 1-2-4 & 3-5-8 & 6-7-9 | 153. 1-4-7 & 2-3-8 & 5-6-9
14. 1-2-4 & 3-5-9 & 6-7-8 | 154. 1-4-7 & 2-3-9 & 5-6-8
15. 1-2-4 & 3-6-7 & 5-8-9 | 155. 1-4-7 & 2-5-6 & 3-8-9
16. 1-2-4 & 3-6-8 & 5-7-9 | 156. 1-4-7 & 2-5-8 & 3-6-9
17. 1-2-4 & 3-6-9 & 5-7-8 | 157. 1-4-7 & 2-5-9 & 3-6-8
18. 1-2-4 & 3-7-8 & 5-6-9 | 158. 1-4-7 & 2-6-8 & 3-5-9
19. 1-2-4 & 3-7-9 & 5-6-8 | 159. 1-4-7 & 2-6-9 & 3-5-8
20. 1-2-4 & 3-8-9 & 5-6-7 | 160. 1-4-7 & 2-8-9 & 3-5-6
21. 1-2-5 & 3-4-6 & 7-8-9 | 161. 1-4-8 & 2-3-5 & 6-7-9
22. 1-2-5 & 3-4-7 & 6-8-9 | 162. 1-4-8 & 2-3-6 & 5-7-9
23. 1-2-5 & 3-4-8 & 6-7-9 | 163. 1-4-8 & 2-3-7 & 5-6-9
24. 1-2-5 & 3-4-9 & 6-7-8 | 164. 1-4-8 & 2-3-9 & 5-6-7
25. 1-2-5 & 3-6-7 & 4-8-9 | 165. 1-4-8 & 2-5-6 & 3-7-9
26. 1-2-5 & 3-6-8 & 4-7-9 | 166. 1-4-8 & 2-5-7 & 3-6-9
27. 1-2-5 & 3-6-9 & 4-7-8 | 167. 1-4-8 & 2-5-9 & 3-6-7
28. 1-2-5 & 3-7-8 & 4-6-9 | 168. 1-4-8 & 2-6-7 & 3-5-9
29. 1-2-5 & 3-7-9 & 4-6-8 | 169. 1-4-8 & 2-6-9 & 3-5-7
30. 1-2-5 & 3-8-9 & 4-6-7 | 170. 1-4-8 & 2-7-9 & 3-5-6
31. 1-2-6 & 3-4-5 & 7-8-9 | 171. 1-4-9 & 2-3-5 & 6-7-8
32. 1-2-6 & 3-4-7 & 5-8-9 | 172. 1-4-9 & 2-3-6 & 5-7-8
33. 1-2-6 & 3-4-8 & 5-7-9 | 173. 1-4-9 & 2-3-7 & 5-6-8
34. 1-2-6 & 3-4-9 & 5-7-8 | 174. 1-4-9 & 2-3-8 & 5-6-7
35. 1-2-6 & 3-5-7 & 4-8-9 | 175. 1-4-9 & 2-5-6 & 3-7-8
36. 1-2-6 & 3-5-8 & 4-7-9 | 176. 1-4-9 & 2-5-7 & 3-6-8
37. 1-2-6 & 3-5-9 & 4-7-8 | 177. 1-4-9 & 2-5-8 & 3-6-7
38. 1-2-6 & 3-7-8 & 4-5-9 | 178. 1-4-9 & 2-6-7 & 3-5-8
39. 1-2-6 & 3-7-9 & 4-5-8 | 179. 1-4-9 & 2-6-8 & 3-5-7
40. 1-2-6 & 3-8-9 & 4-5-7 | 180. 1-4-9 & 2-7-8 & 3-5-6
41. 1-2-7 & 3-4-5 & 6-8-9 | 181. 1-5-6 & 2-3-4 & 7-8-9
42. 1-2-7 & 3-4-6 & 5-8-9 | 182. 1-5-6 & 2-3-7 & 4-8-9
43. 1-2-7 & 3-4-8 & 5-6-9 | 183. 1-5-6 & 2-3-8 & 4-7-9
44. 1-2-7 & 3-4-9 & 5-6-8 | 184. 1-5-6 & 2-3-9 & 4-7-8
45. 1-2-7 & 3-5-6 & 4-8-9 | 185. 1-5-6 & 2-4-7 & 3-8-9
46. 1-2-7 & 3-5-8 & 4-6-9 | 186. 1-5-6 & 2-4-8 & 3-7-9
47. 1-2-7 & 3-5-9 & 4-6-8 | 187. 1-5-6 & 2-4-9 & 3-7-8
48. 1-2-7 & 3-6-8 & 4-5-9 | 188. 1-5-6 & 2-7-8 & 3-4-9
49. 1-2-7 & 3-6-9 & 4-5-8 | 189. 1-5-6 & 2-7-9 & 3-4-8
50. 1-2-7 & 3-8-9 & 4-5-6 | 190. 1-5-6 & 2-8-9 & 3-4-7
51. 1-2-8 & 3-4-5 & 6-7-9 | 191. 1-5-7 & 2-3-4 & 6-8-9
52. 1-2-8 & 3-4-6 & 5-7-9 | 192. 1-5-7 & 2-3-6 & 4-8-9
53. 1-2-8 & 3-4-7 & 5-6-9 | 193. 1-5-7 & 2-3-8 & 4-6-9
54. 1-2-8 & 3-4-9 & 5-6-7 | 194. 1-5-7 & 2-3-9 & 4-6-8
55. 1-2-8 & 3-5-6 & 4-7-9 | 195. 1-5-7 & 2-4-6 & 3-8-9
56. 1-2-8 & 3-5-7 & 4-6-9 | 196. 1-5-7 & 2-4-8 & 3-6-9
57. 1-2-8 & 3-5-9 & 4-6-7 | 197. 1-5-7 & 2-4-9 & 3-6-8
58. 1-2-8 & 3-6-7 & 4-5-9 | 198. 1-5-7 & 2-6-8 & 3-4-9
59. 1-2-8 & 3-6-9 & 4-5-7 | 199. 1-5-7 & 2-6-9 & 3-4-8
60. 1-2-8 & 3-7-9 & 4-5-6 | 200. 1-5-7 & 2-8-9 & 3-4-6
61. 1-2-9 & 3-4-5 & 6-7-8 | 201. 1-5-8 & 2-3-4 & 6-7-9
62. 1-2-9 & 3-4-6 & 5-7-8 | 202. 1-5-8 & 2-3-6 & 4-7-9
63. 1-2-9 & 3-4-7 & 5-6-8 | 203. 1-5-8 & 2-3-7 & 4-6-9
64. 1-2-9 & 3-4-8 & 5-6-7 | 204. 1-5-8 & 2-3-9 & 4-6-7
65. 1-2-9 & 3-5-6 & 4-7-8 | 205. 1-5-8 & 2-4-6 & 3-7-9
66. 1-2-9 & 3-5-7 & 4-6-8 | 206. 1-5-8 & 2-4-7 & 3-6-9
67. 1-2-9 & 3-5-8 & 4-6-7 | 207. 1-5-8 & 2-4-9 & 3-6-7
68. 1-2-9 & 3-6-7 & 4-5-8 | 208. 1-5-8 & 2-6-7 & 3-4-9
69. 1-2-9 & 3-6-8 & 4-5-7 | 209. 1-5-8 & 2-6-9 & 3-4-7
70. 1-2-9 & 3-7-8 & 4-5-6 | 210. 1-5-8 & 2-7-9 & 3-4-6
71. 1-3-4 & 2-5-6 & 7-8-9 | 211. 1-5-9 & 2-3-4 & 6-7-8
72. 1-3-4 & 2-5-7 & 6-8-9 | 212. 1-5-9 & 2-3-6 & 4-7-8
73. 1-3-4 & 2-5-8 & 6-7-9 | 213. 1-5-9 & 2-3-7 & 4-6-8
74. 1-3-4 & 2-5-9 & 6-7-8 | 214. 1-5-9 & 2-3-8 & 4-6-7
75. 1-3-4 & 2-6-7 & 5-8-9 | 215. 1-5-9 & 2-4-6 & 3-7-8
76. 1-3-4 & 2-6-8 & 5-7-9 | 216. 1-5-9 & 2-4-7 & 3-6-8
77. 1-3-4 & 2-6-9 & 5-7-8 | 217. 1-5-9 & 2-4-8 & 3-6-7
78. 1-3-4 & 2-7-8 & 5-6-9 | 218. 1-5-9 & 2-6-7 & 3-4-8
79. 1-3-4 & 2-7-9 & 5-6-8 | 219. 1-5-9 & 2-6-8 & 3-4-7
80. 1-3-4 & 2-8-9 & 5-6-7 | 220. 1-5-9 & 2-7-8 & 3-4-6
81. 1-3-5 & 2-4-6 & 7-8-9 | 221. 1-6-7 & 2-3-4 & 5-8-9
82. 1-3-5 & 2-4-7 & 6-8-9 | 222. 1-6-7 & 2-3-5 & 4-8-9
83. 1-3-5 & 2-4-8 & 6-7-9 | 223. 1-6-7 & 2-3-8 & 4-5-9
84. 1-3-5 & 2-4-9 & 6-7-8 | 224. 1-6-7 & 2-3-9 & 4-5-8
85. 1-3-5 & 2-6-7 & 4-8-9 | 225. 1-6-7 & 2-4-5 & 3-8-9
86. 1-3-5 & 2-6-8 & 4-7-9 | 226. 1-6-7 & 2-4-8 & 3-5-9
87. 1-3-5 & 2-6-9 & 4-7-8 | 227. 1-6-7 & 2-4-9 & 3-5-8
88. 1-3-5 & 2-7-8 & 4-6-9 | 228. 1-6-7 & 2-5-8 & 3-4-9
89. 1-3-5 & 2-7-9 & 4-6-8 | 229. 1-6-7 & 2-5-9 & 3-4-8
90. 1-3-5 & 2-8-9 & 4-6-7 | 230. 1-6-7 & 2-8-9 & 3-4-5
91. 1-3-6 & 2-4-5 & 7-8-9 | 231. 1-6-8 & 2-3-4 & 5-7-9
92. 1-3-6 & 2-4-7 & 5-8-9 | 232. 1-6-8 & 2-3-5 & 4-7-9
93. 1-3-6 & 2-4-8 & 5-7-9 | 233. 1-6-8 & 2-3-7 & 4-5-9
94. 1-3-6 & 2-4-9 & 5-7-8 | 234. 1-6-8 & 2-3-9 & 4-5-7
95. 1-3-6 & 2-5-7 & 4-8-9 | 235. 1-6-8 & 2-4-5 & 3-7-9
96. 1-3-6 & 2-5-8 & 4-7-9 | 236. 1-6-8 & 2-4-7 & 3-5-9
97. 1-3-6 & 2-5-9 & 4-7-8 | 237. 1-6-8 & 2-4-9 & 3-5-7
98. 1-3-6 & 2-7-8 & 4-5-9 | 238. 1-6-8 & 2-5-7 & 3-4-9
99. 1-3-6 & 2-7-9 & 4-5-8 | 239. 1-6-8 & 2-5-9 & 3-4-7
100. 1-3-6 & 2-8-9 & 4-5-7 | 240. 1-6-8 & 2-7-9 & 3-4-5
101. 1-3-7 & 2-4-5 & 6-8-9 | 241. 1-6-9 & 2-3-4 & 5-7-8
102. 1-3-7 & 2-4-6 & 5-8-9 | 242. 1-6-9 & 2-3-5 & 4-7-8
103. 1-3-7 & 2-4-8 & 5-6-9 | 243. 1-6-9 & 2-3-7 & 4-5-8
104. 1-3-7 & 2-4-9 & 5-6-8 | 244. 1-6-9 & 2-3-8 & 4-5-7
105. 1-3-7 & 2-5-6 & 4-8-9 | 245. 1-6-9 & 2-4-5 & 3-7-8
106. 1-3-7 & 2-5-8 & 4-6-9 | 246. 1-6-9 & 2-4-7 & 3-5-8
107. 1-3-7 & 2-5-9 & 4-6-8 | 247. 1-6-9 & 2-4-8 & 3-5-7
108. 1-3-7 & 2-6-8 & 4-5-9 | 248. 1-6-9 & 2-5-7 & 3-4-8
109. 1-3-7 & 2-6-9 & 4-5-8 | 249. 1-6-9 & 2-5-8 & 3-4-7
110. 1-3-7 & 2-8-9 & 4-5-6 | 250. 1-6-9 & 2-7-8 & 3-4-5
111. 1-3-8 & 2-4-5 & 6-7-9 | 251. 1-7-8 & 2-3-4 & 5-6-9
112. 1-3-8 & 2-4-6 & 5-7-9 | 252. 1-7-8 & 2-3-5 & 4-6-9
113. 1-3-8 & 2-4-7 & 5-6-9 | 253. 1-7-8 & 2-3-6 & 4-5-9
114. 1-3-8 & 2-4-9 & 5-6-7 | 254. 1-7-8 & 2-3-9 & 4-5-6
115. 1-3-8 & 2-5-6 & 4-7-9 | 255. 1-7-8 & 2-4-5 & 3-6-9
116. 1-3-8 & 2-5-7 & 4-6-9 | 256. 1-7-8 & 2-4-6 & 3-5-9
117. 1-3-8 & 2-5-9 & 4-6-7 | 257. 1-7-8 & 2-4-9 & 3-5-6
118. 1-3-8 & 2-6-7 & 4-5-9 | 258. 1-7-8 & 2-5-6 & 3-4-9
119. 1-3-8 & 2-6-9 & 4-5-7 | 259. 1-7-8 & 2-5-9 & 3-4-6
120. 1-3-8 & 2-7-9 & 4-5-6 | 260. 1-7-8 & 2-6-9 & 3-4-5
121. 1-3-9 & 2-4-5 & 6-7-8 | 261. 1-7-9 & 2-3-4 & 5-6-8
122. 1-3-9 & 2-4-6 & 5-7-8 | 262. 1-7-9 & 2-3-5 & 4-6-8
123. 1-3-9 & 2-4-7 & 5-6-8 | 263. 1-7-9 & 2-3-6 & 4-5-8
124. 1-3-9 & 2-4-8 & 5-6-7 | 264. 1-7-9 & 2-3-8 & 4-5-6
125. 1-3-9 & 2-5-6 & 4-7-8 | 265. 1-7-9 & 2-4-5 & 3-6-8
126. 1-3-9 & 2-5-7 & 4-6-8 | 266. 1-7-9 & 2-4-6 & 3-5-8
127. 1-3-9 & 2-5-8 & 4-6-7 | 267. 1-7-9 & 2-4-8 & 3-5-6
128. 1-3-9 & 2-6-7 & 4-5-8 | 268. 1-7-9 & 2-5-6 & 3-4-8
129. 1-3-9 & 2-6-8 & 4-5-7 | 269. 1-7-9 & 2-5-8 & 3-4-6
130. 1-3-9 & 2-7-8 & 4-5-6 | 270. 1-7-9 & 2-6-8 & 3-4-5
131. 1-4-5 & 2-3-6 & 7-8-9 | 271. 1-8-9 & 2-3-4 & 5-6-7
132. 1-4-5 & 2-3-7 & 6-8-9 | 272. 1-8-9 & 2-3-5 & 4-6-7
133. 1-4-5 & 2-3-8 & 6-7-9 | 273. 1-8-9 & 2-3-6 & 4-5-7
134. 1-4-5 & 2-3-9 & 6-7-8 | 274. 1-8-9 & 2-3-7 & 4-5-6
135. 1-4-5 & 2-6-7 & 3-8-9 | 275. 1-8-9 & 2-4-5 & 3-6-7
136. 1-4-5 & 2-6-8 & 3-7-9 | 276. 1-8-9 & 2-4-6 & 3-5-7
137. 1-4-5 & 2-6-9 & 3-7-8 | 277. 1-8-9 & 2-4-7 & 3-5-6
138. 1-4-5 & 2-7-8 & 3-6-9 | 278. 1-8-9 & 2-5-6 & 3-4-7
139. 1-4-5 & 2-7-9 & 3-6-8 | 279. 1-8-9 & 2-5-7 & 3-4-6
140. 1-4-5 & 2-8-9 & 3-6-7 | 280. 1-8-9 & 2-6-7 & 3-4-5

 

[sandip.dhamapurkar]

One more thing to clarify things out:

Take the first example (01. 1-2-3 & 4-5-6 & 7-8-9)

There are three combinations of three numbers and it covers all basic
nine numbers
1 2 3
4 5 6
7 8 9

Second example (02. 1-2-3 & 4-5-7 & 6-8-9)

Again there are three combinations of three numbers and it also covers
all the basic nine numbers but BOTH AS A SET are different from each
other.
1 2 3
4 5 7 (the number 7 was in third combination in first example)
6 8 9 (the number 6 was in second combination in first example)

I hope this will be helpful to understand the whole scenario.

 

[Tom Ogilvy]

Why isn't
1-6-8 & 3-5-7 & 2-4-9

a legal combination.

 

[sandip.dhamapurkar]

Because it is already there in the list of 280. Have a look at 237th
record 1-6-8 & 2-4-9 & 3-5-7

The basic idea is that there should be 3 combinations of 3 numbers each
and all the 9 basic numbers should be covered.

The formula that I discoved is as follows but I am not sure if this is
mathematically correct.

Basic numbers : 9
Combinations: 3

9/3=3 (there should be three steps to arrive to the final answer)

step1:

=combin(basic numbers,combination)/3
=combin(9,3)/3
=84/3=28

step2:
=combin(basic numbers - 3,combinations)/2
=combin(6,3)/2
=20/2=10

step3:
=combin(basic numbers - 6,combinations)/1
=combin(3,3)/1
=1/1=1

step1 answer x step2 answer x step3 answer

28*10*1=280