- With 16 blank flags, 33 patterns of 16 colors each (528 uniquely-colored patterns), and 0 to 6 patterns per flag, the number of uniquely crafted banners is 16 × ( 5280 + 5281 + 5282 + 5283 + 5284 + 5285 + 5286 ) ≈ 347 quadrillion. The number of visually distinct flags is smaller, because patterns may completely cover other patterns.
So if this is correct the actual calculation should be:
(Carrots denote powers raised to another power because superscripts cannot be added onto other superscripts.)
16 x (528(0^2) + 528(1^2) + 528(2^2) + 528(3^2) + 528(4^2) + 528(5^2) + 528(6^2)) = 1.6555x1099 different combinations
The powers are squared because for every "n" number of patterns there are added, there are "n" number of "slots" in the crafting order each of those patterns could be placed to make another unique banner. Simplified it is:
16 x (5280 + 5281 + 5284 + 5289 + 52816 + 52825 + 52836) = 1.6555x1099 different combinations
However, this does result in many more banners that are not visually distinct. This is because when adding patterns of the same color, order does not matter one bit and the banners will look identical except when the cursor is hovered over the item in the inventory to view the list of patterns. The order does however, still matter when patterns of different colors are added.
Please discuss and correct me if I missed something that would make this entire calculation bogus.