This question was asked ages ago. I'm a much wiser person now, so I'm gonna throw my two cents' hat in the ring.
First of all, let's address the the additive vs subtractive (or light vs pigment) debate that many answerers had expertly explained in various highly unassuming, specific and instructive ways. Hopefully, once and for all.
The whole deal with mixing paints or inks is that they contain pigments, which absorb light. The simple reason that a red ink appears red is because that red ink absorbs mostly all colored light waves, except the red ones (and yes, "white" light isn't white, it's ALL the colors). When you shine white light on a red ink, it absorbs all the "hidden" waves that are green, blue, purple, etc., and reflects only the red ones, and all that reflected red reaches your eye, and you see the ink as red. That's just how pigments work.
As for light, you just mix different red, green, blue wavelengths together. Neither absorption nor reflection need be involved. Mixing light is "additive mixing". Mixing pigments is "subtractive-additive" (subtractive mixing because of absorption, "subtracting" from white light, and additive mixing happens during reflection)
A major implication of this fact is that, if you mix two or more inks together, you might end up with no reflected light at all. Think about it, if a red ink only reflects red light, and a yellow ink only reflects yellow light, what happens when they're mixed together? Black, of course. The red ink will absorb all the yellow reflected by the yellow ink, and the yellow ink will absorb all the red reflected by the red ink. So there's nothing reflected at all, and what you get is black, the absence of light.
But to this point, you might correctly respond, "OK, smarty pants, if that's how it works, then how come I still got orange by mixing red ink and yellow ink?" Fair point, because what I mentioned above is, ideally, how it should work. But the real world is very messy. That means the idea of a red ink that only reflects red light is nothing but a preposterous fantasy. So is the idea of a yellow ink that only reflects yellow light.
In reality, you rarely ever get anything approaching black. The best you can get is a dark brownish color by mixing red ink and green ink, or a dark purplish color by mixing red ink and blue ink. But still, why does red plus yellow equal orange, and oftentimes, a rather bright orange at that? The answer lies in the overlap between the reflectance range of red inks and yellow inks. Let's take a look at a simplified reflectance graph of various pigments used to make inks, paints, dyes, etc. Reflectance, or the amount of specific colored light reflected by a pigment, was measured relatively to a sample white pigment weighted as 100%.
(Graph made with the program Graph, with data from chsopensource.org)
Clearly it's never black and white that one pigment only reflects one type of light. There's huge overlap between cadmium red and cadmium yellow. Turns out, despite their appearances, both these pigments reflect quite a lot of wavelengths in the red-orange area as well, which explains why you get orange by mixing red and yellow PIGMENTS.
So, if pigments actually have ranges of colors to reflect, for a pigment to be a good mixer, the key is bandwidth. The broader a pigment's bandwidth is, the more different colors it can reflect, the better it can mix with other pigments as losslessly as possible.
For this reason, red, green and blue pigments are the worst choices imaginable. See, not all colors on the visible spectrum are created equal. The three so-called primaries, take up huge chunks of it. Check out this Wikipedia article for more detail, but the gist is that red, green and blue are highly broadband colors, taking up bands of upwards of 65 nm.
Here's where I have some bad news, I can't explain exactly why. But it does seem there's somewhat of a limit to how broad a pigment's bandwidth can be. If you could just imagine that a given physical pigment can only reflect so many different wavelengths, then if a color occupies too many wavelengths, there won't be enough room for other colors, for that pigment to reflect. A red ink and a green ink would make a horrible brownish mix, not because red is the "complementary" of green or some nonsense like that, but because there's simply no red and green pigments in existence with bandwidth large enough to overlap. If there were, you would probably end up with a yellowish mix, the same way you mix a red and a green light together. Likewise, a mixture of red and blue pigments would be horrendous, not only because it's basically impossible for them to overlap in any acceptable way, but because blue pigments are inherently dark (which is inherent to the color blue itself, it's just a dark color). Looking back at the above graph, you can clearly see ultramarine blue's pitiful reflectance.
Meanwhile, compared to red, green and blue, the bandwidths of cyan, magenta and yellow are much more favorable for how narrow they are (downwards of 25 nm). So there's more room for other colors, so to speak. Magenta, or any purplish colors for that matter, occur in excellent mixing pigments because their bandwidth is literally 0 nm. Any magenta or purplish pigment has reflectance coverage of colors in both the short end (violet-blue) and the long end (red-orange) on the visible spectrum. Cyan pigments cover both green and blue, and yellow pigments cover both red and green. This is why, among the three so-called "artist's primaries," yellow carries the entire team, while red and blue have to rely on their respectively "cool red" (basically magenta) and "warm blue" (basically cyan) neighbors on the spectrum.
Despite their shortcomings, red and blue inks are still found in high-end photo printers, for their importance. A red mixed from yellow and magenta is always inferior to a genuine-article red. On the other hand, it does seem challenging to produce a satisfying magenta ink since their typical variety often result in an overly purplish red.
So long answer short, printers only mix three inks to cover as many colors as possible. To do so, they necessarily need pigments that have broad coverage of different colors, namely yellows, cyans, and especially magentas. Red, green and blue pigments are physically incapable of providing such coverage.
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