If you have a set S with n elements then the binomial coefficients nCk represents the number of all subsets of S that have k elements. You can model this using a jar containing some marbles. Start with a jar with 3 marbles labeled 1,2,3 and you draw 2 marbles, your possible outcomes are {1,2}, {1,3} and {2,3} which are the subsets of {1,2,3} with 2 elements.

The powerset of S is the set of all subsets of S and has 2^(n) elements. In the same way this can be modeled as all possible outcomes from drawing any number of marbles from a jar with n marbles. So the sum of the binomial coefficients (nC0+nC1+...+nCn) is 2^(n).

In your example of flipping a coin, (p+q)^(n) each term will be nCk p^(k)q^(n-k) and if p=q=1/2 then p^(k)q^(n-k)=1/2^(n) for each k and

(p+q)^(n)=(1/2)^(n)[nC0 + nC1 + ... + nCn] = 1