Number of induced graphs

Question

How many induced graphs has a graph with $n$ vertices?

I think that there are $2^n=\sum_{k=0}^n \binom{n}{k}$.Is this correct?

Answer 1

If the original vertex set is $V$ then any subset of $K\subseteq V$ uniquely determines an induced graph. Hence, the number of induced graphs is equal to the cardinality of the power set $P(V)$ of $V$. Do you see an easy way to show that $|P(V)| = 2^{|V|}$ without applying the binomial theorem?