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What does the sum of all binomial coefficients mean?
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The sum of all binomial coefficients represents the total number of possible outcomes. For n coin flips, there are 2^n possible sequences of heads and tails, which is what the corresponding binomial coefficients add up to.
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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
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Consider  (1 + 1)\^n. Expanding this shows that the sum of the binomial coefficients in the formula is equal to 2\^n. For example, for n = 2 you get 1 + 2 + 1 = 2\^2. For n=3 you get 1 + 3 + 3 + 1 = 2\^3.

Taking the case n=3 as an example, the sum represents the number of possible subsets of 3 items, i.e. the number of subsets of size 0 plus the number of size 1, plus the number of size 2, plus the number of size 3...or 1 + 3 + 3 + 1. It makes sense that this should be 2\^3 since each of the 3 items can either be in a given subset or not, so the total number of possibilities is 2 \* 2 \* 2.

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