Binomial identity via IE

Universe. All subsets of $\{1,\ldots,n\}$.

Sets $A_i$ = subsets containing $i$. We want subsets with no element, i.e. $\varnothing$.

IE. $|\varnothing|=2^n-\sum_i|A_i|+\sum_{i<j}|A_i\cap A_j|-\cdots$

$=2^n-n\cdot2^{n-1}+\binom{n}{2}2^{n-2}-\cdots=\sum_k(-1)^k\binom{n}{k}2^{n-k}$.

Setting this equal to $1$ (the empty set count) and dividing: $\sum_k(-1)^k\binom{n}{k}2^{-k}=2^{-n}$, so multiplying by $2^n$: $\sum_k(-1)^k\binom{n}{k}2^{n-k}=1$, i.e. $\sum_k(-1)^k\binom{n}{k}=0$.

Conclusion. The identity holds.