If you only want to use the pre-defined colors:
27 face decals * 26 colors * 28 chest decals * 26 colors * 2 chests * 26 colors * 2 waists * 26 colors * 7 hats * 26 colors * 6 backpacks * 26 colors * 2 shoulders * 26 colors * 2 hands * 26 colors * 2 feet * 26 colors * 4 accents * 35 colors * 6 shoulder pads * 26 colors * 2 shoulders * 26 colors * 2 hands * 26 colors * 2 feet * 26 colors = 16,941,187,959,083,683,010,398,126,080 septillion (short)/quadrillion (long) permutations ~= 1.69e+24
If you want to use any RGB color:
27 face decals * 1000000000 colors * 28 chest decals * 1000000000 colors * 2 chests * 1000000000 colors * 2 waists * 1000000000 colors * 7 hats * 1000000000 colors * 6 backpacks * 1000000000 colors * 2 shoulders * 1000000000 colors * 2 hands * 1000000000 colors * 2 feet * 1000000000 colors * 4 accents * 1000000000000 colors * 6 shoulder pads * 1000000000 colors * 2 shoulders * 1000000000 colors * 2 hands * 1000000000 colors * 2 feet * 1000000000 colors ~= 1.95e+137 permutations
A few problems though. Having no hat, backpack, shoulder pad, or accent leads to the colors not affecting the player visually. You also can't use the visor with some hats without scripts. From a code point of view the number is correct but for player use the actual number is a bit off but this is close enough.