Counting how many states can be specified by 16 bits

Start with all 16 bits (represented here by white or black pixels) white.

There are two ways of filling only the first pixel (in reading order), either color it black or leave it white:
fill first of 16 squares in two ways

N = 2

There are four ways of filling the second pixel given the first pixel's possible states, two for each of the two above:
fill second of 16 squares in two ways

N = 4

    Repeat for all 16 pixels:
   3rd ...N = 8
   5th ...N = 16
   6th ...N = 32
   7th ...N = 64
   8th ...N = 128
   9th ...N = 256
   10th...N = 512
   11th...N = 1024
   12th...N = 2048
   13th...N = 4096
   14th...N = 8192
   15th...N = 16384
   16th...N = 32768

    This procedure results in every possible 4 x 4, 1 bit image. Adding all the N's together results in 65536 = 2¹⁶ possible states.