TITLE: Fibonacci combinatorial identity: $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$
QUESTION [5 upvotes]: Can someone explain how to prove the following identity involving Fibonacci sequence 
$F_n$
$F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$
?

REPLY [2 votes]: Recall that $F_{2n}$ is the number of ways to tile a $1\times (2n+1)$ rectangle with squares and dominoes.
Let's focus on first $n$ tiles. We can choose which of them are squares in $\binom ni$ ways and the rest $(2n+1)-i-2(n-i)=i+1$ cells can be tiled in $F_i$ ways.