The Ultimate Question of Life, the Universe, and Everything

Moderators, please pin this in important!

So, here is the formulation of that very question.

The answer is very simple. Such polls should provide maximum info about HWers' tastes. To get maximum info, the number of answer options in the poll should be as large as possible. So, it's time to turn to combinatorics, namely the concept of combinations!

In combinatorics, a combination from n {\displaystyle n} by k {\displaystyle k} is a set of k {\displaystyle k} elements chosen from a given set containing n {\displaystyle n} distinct elements.

Well, that's not hard, it's not Newton's binomial... Oh wait, it actually is exactly that.

The number of combinations from n {\displaystyle n} by k {\displaystyle k} equals the binomial coefficient

This thing shows us how many different answers can be given in a poll where n is the number of items in the poll, and k is how many checkboxes at once the poll author has allowed us to tick.
It is precisely the number of combinations that we need to maximize so that the poll brings the greatest amount of information about forum members.
Generally speaking, it becomes quite obvious that for this, k needs to be chosen as something in between.

Pascal's triangle hints at this

Then the optimal number of how many answers one can choose seems like this:

Number of answer options in the poll = n
How many you can pick at once = k

n k
2 1
3 1
4 2
5 2
6 3
7 3
8 4
9 4
10 5
11 5
12 6
13 6
14 7
......

In other words, divide n in half and round either way. I chose rounding down because changing this number downward seems most reasonable (it's easier to choose).

Actually, when should one deviate from this rule in favor of smaller k? The answer is also obvious — if you believe that the number n is actually exaggerated.

A simple example: my Arknights poll contained 15 answer options, but two of them were guys. This means that guys entering the thread definitely won't choose those answer options. Well, girls entering the thread, on the contrary, will tend to choose the guys, while they are more indifferent to the girls. Based on this, the estimated value of n is 13, not 15. The estimated value of k in that case = 6.

Should I have then allowed choosing up to 6 answer options in the poll?

Well, perhaps still no. An additional consideration (though less clear-cut and mathematical) could be the estimated quality of the content. If you look at the girls in the poll and realize that yes, they're all just top-notch super-top-tier, then you should strictly follow the instruction and take k=n/2. Otherwise, you need to realize that some HWers won't want to pick so many answer options and will either deliberately choose fewer (ruining your statistics), or will suffer the agony of choice among several bad (for them) options. And our goal is not to torment anyone.

Based on this, when creating a poll, you need to subjectively determine an additional coefficient m, taking it within the range 0 < m <= 1.
In my Arknights poll, such a coefficient can be pulled out of thin air as 0.5 — and get exactly the choice of 3 out of 15.

That is, the final formula

k = n * m / 2

Note that if your m is too small, then either you're estimating it incorrectly, or there's something wrong with your content (why make a poll about insufficiently good girls?).

Thanks for your attention. Feel free to ask questions.