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).