My friend is designing a logo. The logo can essentially be reduced to 24 coloured dots arranged in a circle, and they can be either red or white. We want to produce a individual variation of this logo for each employee. That, if I have worked it out right, (since this appears analogous to a 24-bit binary string), means we could have an individual logo for 2^24 employees, obviously way more than we need.
But of course, we don't really want logos that don't have a lot of white dots as they may look too sparse. So we stipulate that there must always be at least half + 1 = 13 in the logo. How many combinations does that restrict us to?
My initial thought is 12 (half) + 1 + 2^11, but I'm not good enough to prove it.
Also, how can we generalise this formula for $x$ dots, $y$ individual colours and at least $n$ colours of a single type? If that's too general, what about just the case $y = 2$ as we have above?