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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

1

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.

${}$

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