Diagonals of Pascal's triangle

Statement

The sum of binomial coefficients along a shallow diagonal of Pascal’s triangle gives a Fibonacci number: $\displaystyle\sum_{k\ge 0}\binom{n-k}{k}=F_{n+1}$. Compute $\binom{6}{0}+\binom{5}{1}+\binom{4}{2}+\binom{3}{3}$.

Solution

$\binom{6}{0}+\binom{5}{1}+\binom{4}{2}+\binom{3}{3}=1+5+6+1=\boxed{13}=F_7.$