A 2Mp camera is close enough to HD to fill a 1080p x 1920 screen without a real problem (which is saying it's enough for the majority of TVs and Computer monitors in the Western World).
Clearly you have almost no option to crop into a 2Mp image without it looking bad very quickly but, if you frame the shot well and don't crop, you can happily print a 10x8* in my experience.
*When viewed from an appropriate distance.
This is a key part of the whole image sharpness/detail equation and there are formulas to calculate how far you should view images from, depending upon the pixel count and the printing resolution (pixels per inch).
Billboards are incredibly low-resolution but appear pin-sharp when viewed from the correct distance.
Get it right in the viewfinder, rather than relying on cropping later, and 7Mp and above seems more than adequate*
* I just looked back at some of my 7Mp Canon S70 images and you can crop quite a lot.
I wonder just how many people actually test the limits of modern camera sensors of 24Mp and greater.
Here's a 7Mp image from my old S70 and a fairly substantial crop into it. The crop would make a perfectly acceptable 10x8 or more(you can zoom into it)
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The drawback with these older cameras isn't the lower pixel count, it's the speed that they tend to operate, poor low-light ability and poor dynamic range.
They force you to get everything spot on...which is no bad thing.
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