A charge-coupled device (CCD) is a computer chip that can be attached to telescopes and "functions as an electronic camera," enabling astronomers to capture images of stars (Astronomy Education at the University of Nebraska-Lincoln, "Properties of CCDs"). CCDs are made up of "highly efficient light detectors that count incoming photons" ("Properties of CCDs"). The term photon is a word we use to refer to a quantum, or a small amount, of light. As each photon hits a pixel on the chip--pixel being a term that simply means location--the chip counts each photon and stores the count on a computer. The photons are detected through electrical charges, and the number of photons on each pixel creates an image.
However, each pixel is only able to count a maximum number of photons. The maximum number of photons a pixel can count is determined by the CCD's bit depth b. If we adjust the bit depth, we can adjust the maximum number of photons the pixels can count. Failure to adjust the bit depth can actually result in what we call "saturation," and the image will appear with "blooming," meaning streaks ("Properties of CCDs"). We can also prevent saturation by adjusting the exposure time.
Your problem starts out with a 12-bit CCD taking an image over the course of a 10-second exposure; the number of photons that hit the central pixel is 1,068. You want to know just how long an exposure you can take without reaching saturation. We can use the CCD simulator on the Astronomy Education at the University of Nebraska-Lincoln website to see that the maximum number of photons that can hit the central pixel with the CCD bit depth at 12 is actually 4,095; therefore, we know we can certainly increase the exposure time a great deal before we reach saturation. The simulator shows us an image of a star and uses red spots to indicate "saturated pixels" ("Properties of CCDs"). As we keep increasing the exposure time in the simulator, we see that the central pixel does not become saturated until the exposure time reaches 40s; therefore, one can expose an image of a star for up to 39 seconds without saturating the central pixel if the CCD bit depth is set at 12.