Show that

$([n],[0])+([n-1],[1])+([n-2],[2])+$cdots$+([1],[n-1])=f_{n+1},$

where $n$ is a nonnegative integer and $f_{n+1}$ is the $(n+1)$ st Fibonacci number. $($See Appendix B

for a review of binomial coefficients.$)$