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Harmonic Harness (Posted on 2013-11-27) Difficulty: 4 of 5
Three positive integers are chosen at random without replacement from 1,2,....,72. What is the probability that the numbers chosen are in harmonic sequence?

Order of choice doesn't matter. For example, 6-3-2 would qualify as numbers in harmonic sequence.

Bonus Question:

Generalize this result (in terms of n) covering the situation where three positive integers are chosen at random without replacement from 1,2,...,n.

No Solution Yet Submitted by K Sengupta    
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Solution computer solution and extension, but not the bonus | Comment 1 of 2

 10   for N=4 to 90
 15    Ct=0:Good=0
 20    for A=1 to N-2
 30      for B=A+1 to N-1
 40        for C=B+1 to N
 45          Ct=Ct+1
 50          if 1//C-1//B=1//B-1//A then Good=Good+1
 55        next
 60      next
 70    next
 75    print N,Good//Ct,Good;Ct,Good/Ct
 80   next
 
produces the table below, from which you can see that for the case n = 72, the probability is 1/1065.

n            probability
       reduced        unreduced            decimal
                      num den=C(n,3)
4       0              0  4            0
5       0              0  10           0
6       1/10           2  20           0.1
7       2/35           2  35           0.0571428571428571428
8       1/28           2  56           0.0357142857142857142
9       1/42           2  84           0.0238095238095238095
10      1/60           2  120          0.0166666666666666666
11      2/165          2  165          0.0121212121212121212
12      1/55           4  220          0.0181818181818181818
13      2/143          4  286          0.013986013986013986
14      1/91           4  364          0.010989010989010989
15      6/455          6  455          0.0131868131868131868
16      3/280          6  560          0.0107142857142857142
17      3/340          6  680          0.0088235294117647058
18      1/102          8  816          0.0098039215686274509
19      8/969          8  969          0.0082559339525283797
20      1/114          10  1140        0.0087719298245614034
21      1/133          10  1330        0.0075187969924812029
22      1/154          10  1540        0.0064935064935064935
23      10/1771        10  1771        0.0056465273856578204
24      3/506          12  2024        0.0059288537549407114
25      3/575          12  2300        0.005217391304347826
26      3/650          12  2600        0.0046153846153846153
27      4/975          12  2925        0.0041025641025641025
28      1/234          14  3276        0.0042735042735042735
29      1/261          14  3654        0.0038314176245210727
30      9/2030         18  4060        0.0044334975369458127
31      18/4495        18  4495        0.0040044493882091211
32      9/2480         18  4960        0.0036290322580645161
33      9/2728         18  5456        0.0032991202346041055
34      9/2992         18  5984        0.0030080213903743315
35      4/1309         20  6545        0.0030557677616501145
36      11/3570        22  7140        0.0030812324929971988
37      11/3885        22  7770        0.0028314028314028313
38      11/4218        22  8436        0.0026078710289236604
39      22/9139        22  9139        0.0024072655651603019
40      3/1235         24  9880        0.0024291497975708501
41      6/2665         24  10660       0.0022514071294559099
42      1/410          28  11480       0.0024390243902439024
43      4/1763         28  12341       0.0022688598979013045
44      1/473          28  13244       0.0021141649048625792
45      16/7095        32  14190       0.0022551092318534178
46      8/3795         32  15180       0.0021080368906455862
47      32/16215       32  16215       0.0019734813444341658
48      17/8648        34  17296       0.0019657724329324698
49      17/9212        34  18424       0.0018454190186712982
50      17/9800        34  19600       0.0017346938775510203
51      2/1225         34  20825       0.0016326530612244897
52      1/650          34  22100       0.0015384615384615384
53      1/689          34  23426       0.0014513788098693759
54      1/689          36  24804       0.0014513788098693759
55      4/2915         36  26235       0.0013722126929674098
56      19/13860       38  27720       0.0013708513708513708
57      1/770          38  29260       0.0012987012987012987
58      1/812          38  30856       0.001231527093596059
59      2/1711         38  32509       0.0011689070718877849
60      11/8555        44  34220       0.0012857977790765633
61      22/17995       44  35990       0.0012225618227285356
62      11/9455        44  37820       0.0011634056054997355
63      46/39711       46  39711       0.0011583692175971393
64      23/20832       46  41664       0.0011040706605222733
65      23/21840       46  43680       0.0010531135531135531
66      5/4576         50  45760       0.0010926573426573426
67      10/9581        50  47905       0.001043732387015969
68      25/25058       50  50116       0.0009976853699417351
69      25/26197       50  52394       0.0009543077451616596
70      13/13685       52  54740       0.0009499451954694921
71      52/57155       52  57155       0.0009098066660834572
72      1/1065         56  59640       0.0009389671361502347
73      14/15549       56  62196       0.0009003794456235127
74      7/8103         56  64824       0.0008638775762063433
75      58/67525       58  67525       0.0008589411329137356
76      29/35150       58  70300       0.0008250355618776671
77      6/7315         60  73150       0.0008202323991797675
78      31/38038       62  76076       0.0008149744991850254
79      62/79079       62  79079       0.0007840261004817966
80      4/5135         64  82160       0.0007789678675754624
81      8/10665        64  85320       0.0007501172058134083
82      4/5535         64  88560       0.0007226738934056007
83      64/91881       64  91881       0.0006965531502704584
84      5/6806         70  95284       0.0007346459006758741
85      1/1411         70  98770       0.0007087172218284904
86      1/1462         70  102340      0.0006839945280437756
87      14/21199       70  105995      0.0006604085098353695
88      9/13717        72  109736      0.0006561201428883866
89      18/28391       72  113564      0.0006340037335775421
90      13/19580       78  117480      0.0006639427987742594

The unreduced form is included in case it might provide a hint as to solving the bonus portion.


  Posted by Charlie on 2013-11-27 16:45:02
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