LESS COMBINATIONS

 Eight bits have 256 combinations, but if you know that eight out of eight bits are zero, than there's just one combination possible: 00000000. Thus knowing how many bits are 0 and 1, can reduce the amount of possible combinations extremely. The more information there is about a data block, the less combinations are possible. Not just knowing how many zeros and ones there are but knowing more; how many groups of each: 00, 01, 10, and 11 there are (or larger groups), reduces the amount of possible combinations even more. Next are all 256 combinations of 8 bits, in order of the ratio 1/0

 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 1 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 1 2 3 4 5 6 7 8 1 00000000 00000001 00000010 00000100 00001000 00010000 00100000 01000000 10000000 00000011 00000101 00000110 00001001 00001010 00001100 00010001 00010010 00010100 00011000 00100001 00100010 00100100 00101000 00110000 01000001 01000010 01000100 01001000 01010000 01100000 10000001 10000010 10000100 10001000 10010000 10100000 11000000 00000111 00001011 00001101 00001110 00010011 00010101 00010110 00011001 00011010 00011100 00100011 00100101 00100110 00101001 00101010 00101100 00110001 00110010 00110100 00111000 01000011 01000101 01000110 01001001 01001010 01001100 01010001 01010010 01010100 01011000 01100001 01100010 01100100 01101000 01110000 10000011 10000101 10000110 10001001 10001010 10001100 10010001 10010010 10010100 10011000 10100001 10100010 10100100 10101000 10110000 11000001 11000010 11000100 11001000 11010000 11100000 00001111 00010111 00011101 00011110 00011011 00100111 00101011 00101101 00101110 00110011 00110101 00110110 00111001 00111010 00111100 01000111 01001011 01001101 01001110 01010011 01010101 01010110 01011001 01011010 01011100 01100011 01100101 01100110 01101001 01101010 01101100 01110001 01110010 01110100 01111000 10000111 10001011 10001101 10001110 10010011 10010101 10010110 10011001 10011010 10011100 10100011 10100101 10100110 10101001 10101010 10101100 10110001 10110010 10110100 10111000 11000011 11000101 11000110 11001001 11001010 11001100 11010001 11010010 11010100 11011000 11100001 11100010 11100100 11101000 11110000 00011111 00101111 00110111 00111011 00111101 00111110 01001111 01010111 01011011 01011101 01011110 01100111 01101011 01101101 01101110 01110011 01110101 01110110 01111001 01111010 01111100 10001111 10010111 10011011 10011101 10011110 10100111 10101011 10101101 10101110 10110011 10110101 10110110 10111001 10111010 10111100 11000111 11001011 11001101 11001110 11010011 11010101 11010110 11011001 11011010 11011100 11100011 11100101 11100110 11101001 11101010 11101100 11110001 11110010 11110100 11111000 00111111 01011111 01101111 01110111 01111011 01111101 01111110 10011111 10101111 10110111 10111011 10111101 10111110 11001111 11010111 11011011 11011101 11011110 11100111 11101011 11101101 11101110 11110011 11110101 11110110 11111001 11111010 11111100 01111111 10111111 11011111 11101111 11110111 11111011 11111101 11111110 11111111 1 combination 8 combinations 28 combinations 56 combinations 70 combinations 56 combinations 28 combinations 8 combinations 1 combination

For eight bits:
 all zero = 1 one = 2 ones = 3 ones = 4 ones = 5 ones = 6 ones = 7 ones = 8 ones = 1 combination 8 combinations 28 combinations 56 combinations 70 combinations 56 combinations 28 combinations 8 combinations 1 combination

 PASCAL'S TRIANGLE

 Hidden in Pascal's Triangle are a lot of useful answers The total of each row is equal to a power of 2 Row nr eight = 1 + 8 + 28 + 56 + 70 + 56 + 28 + 8 + 1 = 256 (equal to 2^8) The formula: Example A How many combinations of 8 bits have: eight 0 and zero 1 P = 8! / (8! * 0!) P = 1 Example B How many combinations of 8 bits have: six 0 and two 1 P = 8! / (6! * 2!) P = 28 Example C How many combinations of 8 bits have: four 0 and four 1 P = 8! / (4! * 4!) P = 70 More than digital How many combinations are possible with more than two tones? Nine tones, represented by nine letters, each appearing once: ABCDEFGHIJ The amount of combinations is: 9! 9! = 9x8x7x6x5x4x3x2x1 = 362880 But when some tones appear more than once, like: AAAABBCCC Than the amount of combinations is less than 9! : 9! / (4! * 2! * 3!) = 1260 8 bits (1 color channel pixel): 1 8 28 56 70 56 28 8 1 24 bits (1 RGB color pixel): 1 24 276 2024 10626 42504 134596 346104 735471 1307504 1961256 2496144 2704156 2496144 1961256 1307504 735471 346104 134596 42504 10626 2024 276 24 1 48 bits (2 RGB color pixels): 1 48 1128 17296 194580 1712304 12271512 73629072 377348994 1677106640 6540715896 22595200368 69668534468 192928249296 482320623240 1093260079344 2254848913647 4244421484512 7309837001104 11541847896480 16735679449896 22314239266528 27385657281648 30957699535776 32247603683100 30957699535776 27385657281648 22314239266528 16735679449896 11541847896480 7309837001104 4244421484512 2254848913647 1093260079344 482320623240 192928249296 69668534468 22595200368 6540715896 1677106640 377348994 73629072 12271512 1712304 194580 17296 1128 48 1 A source: Interactive Pascal's Triangle Special Thanks to Alexandre Iervolino

 Giesbert Nijhuis