How many combinations of coloured dots (with restrictions)? 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?

There are $\binom{24}{12}$ $12$-element subsets of the $24$ positions; those correspond to the designs that have $12$ dots of each color. Half of the remaining $2^{24}-\binom{24}{12}$ subsets have at least $13$ elements. Thus, the answer to the main question is
$$\frac12\left(2^{24}-\binom{24}{12}\right)=2^{23}-\frac12\binom{24}{12}=16,777,216-1,352,078=15,425,138$$
logos. The general problem is significantly harder: for this one we can take advantage of the fact that we’re looking at exactly half of an easily computed number. However, for two colors and a majority of white dots it’s not bad. If there are $2n$ dots, the same reasoning leads to the result
$$\frac12\left(2^{2n}-\binom{2n}n\right)=2^{2n-1}-\frac12\binom{2n}n\;.$$
If there are $2n+1$ dots, exactly half of the $2^{2n+1}$, or $2^{2n}$ designs, have a majority of white dots.
Added: Note that I am assuming, as you implicitly did in saying that there are $2^{24}$ unrestricted patterns, that identical patterns in different orientations are counted as different patterns (e.g., that the $24$ patterns with a single red dot count as $24$ patterns, not as one). The problem becomes much more complicated if patterns that can be rotated into one another are considered identical.
