+753 The numbers 1,2,3,4,5,6,7,8,9 are arranged in a list so that each number is either greater than all the numbers that come before it or is less than all the numbers that come before it. For example, 4,5,6,3,2,7,1,8,9 is one such list: notice that (for instance) the 6 is greater than all the numbers that come before it, and the 2 is less than all the numbers that come before it. How many are such lists of the numbers 1,2,3,4,5,6,7,8,9 possible? Thanks in advance!
+2489 The last digit has to be either a one (1) or nine (9), then are two (2) options (>,<) for the remaining eight (8) digits, so there are 2^8 =256 arraignments where each digit is either greater than or less than the preceding digits.
GA
+26393 The numbers 1,2,3,4,5,6,7,8,9 are arranged in a list so that each number
is either greater than all the numbers that come before it or
is less than all the numbers that come before it.
For example, 4,5,6,3,2,7,1,8,9 is one such list:
notice that (for instance) the 6 is greater than all the numbers that come before it,
and the 2 is less than all the numbers that come before it.
How many are such lists of the numbers 1,2,3,4,5,6,7,8,9 possible?
\(\begin{array}{|rrrr|} \hline \begin{array}{r|r} 1.& 123456789 \\ 2.& 213456789 \\ 3.& 231456789 \\ 4.& 234156789 \\ 5.& 234516789 \\ 6.& 234561789 \\ 7.& 234567189 \\ 8.& 234567819 \\ 9.& 234567891 \\ 10.& 321456789 \\ 11.& 324156789 \\ 12.& 324516789 \\ 13.& 324561789 \\ 14.& 324567189 \\ 15.& 324567819 \\ 16.& 324567891 \\ 17.& 342156789 \\ 18.& 342516789 \\ 19.& 342561789 \\ 20.& 342567189 \\ 21.& 342567819 \\ 22.& 342567891 \\ 23.& 345216789 \\ 24.& 345261789 \\ 25.& 345267189 \\ 26.& 345267819 \\ 27.& 345267891 \\ 28.& 345621789 \\ 29.& 345627189 \\ 30.& 345627819 \\ 31.& 345627891 \\ 32.& 345672189 \\ 33.& 345672819 \\ 34.& 345672891 \\ 35.& 345678219 \\ 36.& 345678291 \\ 37.& 345678921 \\ 38.& 432156789 \\ 39.& 432516789 \\ 40.& 432561789 \\ 41.& 432567189 \\ 42.& 432567819 \\ 43.& 432567891 \\ 44.& 435216789 \\ 45.& 435261789 \\ 46.& 435267189 \\ 47.& 435267819 \\ 48.& 435267891 \\ 49.& 435621789 \\ 50.& 435627189 \\ 51.& 435627819 \\ 52.& 435627891 \\ 53.& 435672189 \\ 54.& 435672819 \\ 55.& 435672891 \\ 56.& 435678219 \\ 57.& 435678291 \\ 58.& 435678921 \\ 59.& 453216789 \\ 60.& 453261789 \\ 61.& 453267189 \\ 62.& 453267819 \\ 63.& 453267891 \\ 64.& 453621789 \\ \end{array} & \begin{array}{r|r} 65.& 453627189 \\ 66.& 453627819 \\ 67.& 453627891 \\ 68.& 453672189 \\ 69.& 453672819 \\ 70.& 453672891 \\ 71.& 453678219 \\ 72.& 453678291 \\ 73.& 453678921 \\ 74.& 456321789 \\ 75.& 456327189 \\ 76.& 456327819 \\ 77.& 456327891 \\ 78.& 456372189 \\ 79.& 456372819 \\ 80.& 456372891 \\ 81.& 456378219 \\ 82.& 456378291 \\ 83.& 456378921 \\ 84.& 456732189 \\ 85.& 456732819 \\ 86.& 456732891 \\ 87.& 456738219 \\ 88.& 456738291 \\ 89.& 456738921 \\ 90.& 456783219 \\ 91.& 456783291 \\ 92.& 456783921 \\ 93.& 456789321 \\ 94.& 543216789 \\ 95.& 543261789 \\ 96.& 543267189 \\ 97.& 543267819 \\ 98.& 543267891 \\ 99.& 543621789 \\ 100.& 543627189 \\ 101.& 543627819 \\ 102.& 543627891 \\ 103.& 543672189 \\ 104.& 543672819 \\ 105.& 543672891 \\ 106.& 543678219 \\ 107.& 543678291 \\ 108.& 543678921 \\ 109.& 546321789 \\ 110.& 546327189 \\ 111.& 546327819 \\ 112.& 546327891 \\ 113.& 546372189 \\ 114.& 546372819 \\ 115.& 546372891 \\ 116.& 546378219 \\ 117.& 546378291 \\ 118.& 546378921 \\ 119.& 546732189 \\ 120.& 546732819 \\ 121.& 546732891 \\ 122.& 546738219 \\ 123.& 546738291 \\ 124.& 546738921 \\ 125.& 546783219 \\ 126.& 546783291 \\ 127.& 546783921 \\ 128.& 546789321 \\ \end{array} & \begin{array}{r|r} 129.& 564321789 \\ 130.& 564327189 \\ 131.& 564327819 \\ 132.& 564327891 \\ 133.& 564372189 \\ 134.& 564372819 \\ 135.& 564372891 \\ 136.& 564378219 \\ 137.& 564378291 \\ 138.& 564378921 \\ 139.& 564732189 \\ 140.& 564732819 \\ 141.& 564732891 \\ 142.& 564738219 \\ 143.& 564738291 \\ 144.& 564738921 \\ 145.& 564783219 \\ 146.& 564783291 \\ 147.& 564783921 \\ 148.& 564789321 \\ 149.& 567432189 \\ 150.& 567432819 \\ 151.& 567432891 \\ 152.& 567438219 \\ 153.& 567438291 \\ 154.& 567438921 \\ 155.& 567483219 \\ 156.& 567483291 \\ 157.& 567483921 \\ 158.& 567489321 \\ 159.& 567843219 \\ 160.& 567843291 \\ 161.& 567843921 \\ 162.& 567849321 \\ 163.& 567894321 \\ 164.& 654321789 \\ 165.& 654327189 \\ 166.& 654327819 \\ 167.& 654327891 \\ 168.& 654372189 \\ 169.& 654372819 \\ 170.& 654372891 \\ 171.& 654378219 \\ 172.& 654378291 \\ 173.& 654378921 \\ 174.& 654732189 \\ 175.& 654732819 \\ 176.& 654732891 \\ 177.& 654738219 \\ 178.& 654738291 \\ 179.& 654738921 \\ 180.& 654783219 \\ 181.& 654783291 \\ 182.& 654783921 \\ 183.& 654789321 \\ 184.& 657432189 \\ 185.& 657432819 \\ 186.& 657432891 \\ 187.& 657438219 \\ 188.& 657438291 \\ 189.& 657438921 \\ 190.& 657483219 \\ 191.& 657483291 \\ 192.& 657483921 \\ \end{array} & \begin{array}{r|r} 193.& 657489321 \\ 194.& 657843219 \\ 195.& 657843291 \\ 196.& 657843921 \\ 197.& 657849321 \\ 198.& 657894321 \\ 199.& 675432189 \\ 200.& 675432819 \\ 201.& 675432891 \\ 202.& 675438219 \\ 203.& 675438291 \\ 204.& 675438921 \\ 205.& 675483219 \\ 206.& 675483291 \\ 207.& 675483921 \\ 208.& 675489321 \\ 209.& 675843219 \\ 210.& 675843291 \\ 211.& 675843921 \\ 212.& 675849321 \\ 213.& 675894321 \\ 214.& 678543219 \\ 215.& 678543291 \\ 216.& 678543921 \\ 217.& 678549321 \\ 218.& 678594321 \\ 219.& 678954321 \\ 220.& 765432189 \\ 221.& 765432819 \\ 222.& 765432891 \\ 223.& 765438219 \\ 224.& 765438291 \\ 225.& 765438921 \\ 226.& 765483219 \\ 227.& 765483291 \\ 228.& 765483921 \\ 229.& 765489321 \\ 230.& 765843219 \\ 231.& 765843291 \\ 232.& 765843921 \\ 233.& 765849321 \\ 234.& 765894321 \\ 235.& 768543219 \\ 236.& 768543291 \\ 237.& 768543921 \\ 238.& 768549321 \\ 239.& 768594321 \\ 240.& 768954321 \\ 241.& 786543219 \\ 242.& 786543291 \\ 243.& 786543921 \\ 244.& 786549321 \\ 245.& 786594321 \\ 246.& 786954321 \\ 247.& 789654321 \\ 248.& 876543219 \\ 249.& 876543291 \\ 250.& 876543921 \\ 251.& 876549321 \\ 252.& 876594321 \\ 253.& 876954321 \\ 254.& 879654321 \\ 255.& 897654321 \\ 256.& 987654321 \\ \end{array}\\ \hline \end{array}\)
The answer is 256
Numbers beginning with 1 = 8C0 =1
Numbers beginning with 2 = 8C1 = 8
Numbers beginning with 3 = 8C2 = 28
Numbers beginning with 4 = 8C3 = 56
Numbers beginning with 5 = 8C4 = 70
Numbers beginning with 6 = 8C5 = 56
Numbers beginning with 7 = 8C6 = 28
Numbers beginning with 8 = 8C7 = 8
Numbers beginning with 9 = 8C8 = 1
Total of all numbers listed by "heureka"(count them!) =1+8+28+56+70+56+28+8+1 = 256 numbers. 2^8 =256 is purely coincidental !!!!!!!!!!!. This is the solution in book!!!!.
+2489 Well, Mr. BB, like usual... you’re an arrogant idiot.
Heureka’s post details the 256 individual solutions to this problem, and they link unambiguously and decidedly to the count obtained via the set theory solution—that is, they are NOT incidental nor are they coincidental. The counts relate to the binomial distribution in Pascal’s triangle. (See below.)
GetSmart is slacking on his job. He should have taught you what a coincidence is. A coincidence is a remarkable concurrence of events or circumstances corresponding in nature or in time of occurrence without an underlying connection.
I should have had Chimp Loki give GetSmart some turkey teeth. That way he could bite you on the áss when you stick your beak in to a math question to cluck and post stúpid and dumb –not to forget rude.
GA
+2489 MIRB, The counts (not the actual sequence) correspond to row #8 of Pascal’s triangle (with zero starting the count, so it’s actually the ninth row). If zero were included in the count then row #9 would give the counts for each number.
Again, despite Mr. BB’s insistence, there is nothing coincidental about these numeric relationships. All of these are mathematically related. The set theory gives the solution to the sum, which is what the question asks for. The binomial distribution gives the leading coefficient for the count of each sequence (via a generating function). Still, further logic and mathematics are required to produce the numeric sequences. There are 362,880 ways to arrange those numbers, but only 256 of them fall within the criteria of the question.
GA
, how long did it take for you to list them out?
