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).