Mistake: formula \(\binomn-k+1k\) is for placing \(k\) non-adjacent indistinct items? No — standard formula for number of binary strings of length \(n\) with no two consecutive 1’s is \(F_n+2\), where \(F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, F_6=8, F_7=13, F_8=21\) - Groen Casting