Help required to compute logic of an answer.

Question

How is the sum of combination series $${20 \choose 1} + {20\choose 2} + {20 \choose 3} +\cdots +{20 \choose 20} = 2^{20}?$$ No one told me or perhaps I missed the logic behind using so in my question.

Answer 1

Considering the expansion , $$(1+x)^{20}={20 \choose 0} + {20\choose 1}x + {20 \choose 2}x^2 +\cdots +{20 \choose 20}x^{20} $$

Plugging in $x=1$, we get, $$2^{20}={20 \choose 0} + {20\choose 1} + {20 \choose 2} +\cdots +{20 \choose 20} $$ $$\implies {20\choose 1} + {20 \choose 2} +\cdots +{20 \choose 20} =2^{20}-1$$

Answer 2

You forgot one term: 20c0 in your notation.

A set having $20$ elements has in total $2^{20}$ subsets.

It has $\binom{20}{k}$ subsets with cardinality $k$ so that: $$2^{20}=\sum_{k=0}^{20}\binom{20}{k}$$