A cereal manufacturer is evaluating a new design for the box of its leading brand. Ten people are shown the new and old box design. Each is asked which one he or she likes best. If each person doesn't have a preference and simply selects one of the two designs at random, what is the probability of each of all 10 selecting the new design?

The answer is 1/1024.

Why is the total number of combination possibilities $\displaystyle 2^{10}\text{?}$