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I am having some difficulty trying to understand how to do this problem. Any help would be great. I'm new to this site. Thanks in advance.

Count the number of in-equivalent arrangements of 1) six red and four black beads on a ring 2) five red, three black, and two yellow beads on a ring.

1 Answers 1

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Judging from the (now deleted) comments, "inequivalent" means inequivalent under flips and rotations. There's three ways I can think of to approach this problem.

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g:=DihedralGroup(IsPermGroup,20);; A:=OrbitsDomain(g,Arrangements([0,0,0,0,0,0,1,1,1,1],10),Permuted);; Size(A); 

returns 16 and

g:=DihedralGroup(IsPermGroup,20);; A:=OrbitsDomain(g,Arrangements([0,0,0,0,0,1,1,1,2,2],10),Permuted);; Size(A); 

returns 132. This is the "easy way".

  • The third way is careful bookkeeping by hand (although, my personal opinion is that the computer will provide a much more accurate answer much faster than hand calculations, so I think this is a waste of time).

Here's an exhaustive enumeration for the first case:

1 rrrrrrbbbb brrrrrrbbb rbbbbrrrrr bbrrrrrrbb bbbbrrrrrr rrbbbbrrrr bbbrrrrrrb rrrbbbbrrr rrrrrbbbbr rrrrbbbbrr  2 rrrrrbrbbb brrrrrbrbb rbbbrbrrrr bbrrrrrbrb bbbrbrrrrr rrbbbrbrrr bbbrrrrrbr bbrbrrrrrb rrrbbbrbrr rrrrbrbbbr rbbbrrrrrb brbrrrrrbb rrrrbbbrbr rrrbrbbbrr brbbbrrrrr rbrrrrrbbb rrrrrbbbrb rrbrbbbrrr rbrbbbrrrr brrrrrbbbr  3 rrrrrbbrbb brrrrrbbrb rbbrbbrrrr bbrrrrrbbr bbrbbrrrrr rrbbrbbrrr rbbrrrrrbb brbbrrrrrb rrrbbrbbrr rrrrbbrbbr  4 rrrrbrrbbb brrrrbrrbb rbbbrrbrrr bbrrrrbrrb bbbrrbrrrr rrbbbrrbrr bbbrrrrbrr bbrrbrrrrb rrrbbbrrbr rrrbrrbbbr rbbbrrrrbr brrbrrrrbb rrrrbbbrrb rrbrrbbbrr rrbbbrrrrb rrbrrrrbbb brrrrbbbrr rbrrbbbrrr brrbbbrrrr rbrrrrbbbr  5 rrrrbrbrbb brrrrbrbrb rbbrbrbrrr bbrrrrbrbr bbrbrbrrrr rrbbrbrbrr rbbrrrrbrb brbrbrrrrb rrrbbrbrbr rrrbrbrbbr brbbrrrrbr rbrbrrrrbb rrrrbbrbrb rrbrbrbbrr rbrbbrrrrb brbrrrrbbr brrrrbbrbr rbrbrbbrrr brbrbbrrrr rbrrrrbbrb  6 rrrrbrbbrb brrrrbrbbr rbrbbrbrrr rbrrrrbrbb brbbrbrrrr rrbrbbrbrr brbrrrrbrb rbbrbrrrrb rrrbrbbrbr bbrbrrrrbr  7 rrrrbbrrbb brrrrbbrrb rbbrrbbrrr bbrrrrbbrr bbrrbbrrrr rrbbrrbbrr rbbrrrrbbr brrbbrrrrb rrrbbrrbbr rrbbrrrrbb  8 rrrbrrrbbb brrrbrrrbb rbbbrrrbrr bbrrrbrrrb bbbrrrbrrr rrbbbrrrbr rrrbbbrrrb rrbrrrbbbr brrrbbbrrr rbrrrbbbrr  9 rrrbrrbrbb brrrbrrbrb rbbrbrrbrr bbrrrbrrbr bbrbrrbrrr rrbbrbrrbr rbbrrrbrrb brbrrbrrrb rrrbbrbrrb rrbrrbrbbr brbbrrrbrr rbrrbrrrbb brrrbbrbrr rbrrbrbbrr rbrbbrrrbr brrbrrrbbr rbrrrbbrbr brrbrbbrrr rrbrbbrrrb rrbrrrbbrb  10 rrrbrrbbrb brrrbrrbbr rbrbbrrbrr rbrrrbrrbb brbbrrbrrr rrbrbbrrbr brbrrrbrrb rbbrrbrrrb rrrbrbbrrb rrbrrbbrbr bbrbrrrbrr bbrrbrrrbr brrrbrbbrr rbrrbbrbrr rbbrbrrrbr brrbrrrbrb rbrrrbrbbr brrbbrbrrr rrbbrbrrrb rrbrrrbrbb  11 rrrbrbrrbb brrrbrbrrb rbbrrbrbrr bbrrrbrbrr bbrrbrbrrr rrbbrrbrbr rbbrrrbrbr brrbrbrrrb rrrbbrrbrb rrbrbrrbbr rrbbrrrbrb rrbrbrrrbb brrrbbrrbr rbrbrrbbrr brrbbrrrbr rbrbrrrbbr rbrrrbbrrb brbrrbbrrr rbrrbbrrrb brbrrrbbrr  12 rrrbrbrbrb brrrbrbrbr rbrbrbrbrr rbrrrbrbrb brbrbrbrrr rrbrbrbrbr brbrrrbrbr rbrbrbrrrb rbrbrrrbrb brbrbrrrbr  13 rrrbbrrrbb brrrbbrrrb rbbrrrbbrr bbrrrbbrrr rrbbrrrbbr  14 rrbrrbrrbb brrbrrbrrb rbbrrbrrbr bbrrbrrbrr rrbbrrbrrb brrbbrrbrr rbrrbrrbbr rbrrbbrrbr brrbrrbbrr rrbrrbbrrb  15 rrbrrbrbrb brrbrrbrbr rbrbrbrrbr rbrrbrrbrb brbrbrrbrr rrbrbrbrrb brbrrbrrbr rbrbrrbrrb brrbrbrbrr rbrrbrbrbr  16 rrbrbrrbrb brrbrbrrbr rbrbrrbrbr rbrrbrbrrb brbrrbrbrr 

and for the second case:

1 rrrrrbbbyy  2 rrrrrbbyby  3 rrrrrbbyyb  4 rrrrrbybby  5 rrrrrbybyb  6 rrrrrybbby  7 rrrrbrbbyy  8 rrrrbrbyby  9 rrrrbrbyyb  10 rrrrbrybby  11 rrrrbrybyb  12 rrrrbryybb  13 rrrrbbrbyy  14 rrrrbbryby  15 rrrrbbryyb  16 rrrrbbbryy  17 rrrrbbbyry  18 rrrrbbyrby  19 rrrrbbyryb  20 rrrrbbybry  21 rrrrbyrbby  22 rrrrbyrbyb  23 rrrrbybrby  24 rrrrbybbry  25 rrrryrbbby  26 rrrrybrbby  27 rrrbrrbbyy  28 rrrbrrbyby  29 rrrbrrbyyb  30 rrrbrrybby  31 rrrbrrybyb  32 rrrbrryybb  33 rrrbrbrbyy  34 rrrbrbryby  35 rrrbrbryyb  36 rrrbrbbryy  37 rrrbrbbyry  38 rrrbrbyrby  39 rrrbrbyryb  40 rrrbrbybry  41 rrrbrbyyrb  42 rrrbryrbby  43 rrrbryrbyb  44 rrrbryrybb  45 rrrbrybrby  46 rrrbrybryb  47 rrrbrybbry  48 rrrbrybyrb  49 rrrbryyrbb  50 rrrbbrrbyy  51 rrrbbrryby  52 rrrbbrryyb  53 rrrbbrbryy  54 rrrbbrbyry  55 rrrbbryrby  56 rrrbbryryb  57 rrrbbrybry  58 rrrbbbrryy  59 rrrbbbryry  60 rrrbbbyrry  61 rrrbbyrrby  62 rrrbbyrryb  63 rrrbbyrbry  64 rrrbbybrry  65 rrrbyrrbby  66 rrrbyrrbyb  67 rrrbyrbrby  68 rrrbyrbryb  69 rrrbyrbbry  70 rrrbybrrby  71 rrrbybrbry  72 rrrbybbrry  73 rrryrrbbby  74 rrryrbrbby  75 rrryrbbrby  76 rrryrbbbry  77 rrrybrrbby  78 rrrybrbrby  79 rrbrrbrbyy  80 rrbrrbryby  81 rrbrrbryyb  82 rrbrrbbryy  83 rrbrrbbyry  84 rrbrrbyrby  85 rrbrrbyryb  86 rrbrrbybry  87 rrbrryrbby  88 rrbrrybrby  89 rrbrbrrbyy  90 rrbrbrryby  91 rrbrbrbryy  92 rrbrbrbyry  93 rrbrbryrby  94 rrbrbryryb  95 rrbrbrybry  96 rrbrbryyrb  97 rrbrbbrryy  98 rrbrbbryry  99 rrbrbbyrry  100 rrbrbyrrby  101 rrbrbyrryb  102 rrbrbyrbry  103 rrbrbyryrb  104 rrbrbybrry  105 rrbryrrbby  106 rrbryrrbyb  107 rrbryrrybb  108 rrbryrbrby  109 rrbryrbryb  110 rrbryrbbry  111 rrbryrbyrb  112 rrbryryrbb  113 rrbrybrrby  114 rrbrybrryb  115 rrbrybrbry  116 rrbrybbrry  117 rrbryyrrbb  118 rrbbrrbyry  119 rrbbrryrby  120 rrbbrbryry  121 rrbbrbyrry  122 rrbbryrrby  123 rrbbryrryb  124 rrbbryrbry  125 rrbbrybrry  126 rrbbbrryry  127 rrbbbryrry  128 rrbyrbrbry  129 rryrbrbrby  130 rryrbrbbry  131 rbrbrbryry  132 rbrbryrbry 

(one representative from each equivalence class, otherwise I'm over the character limit).

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    wow..Thanks so much2012-11-02