Okay so for this one i request to go through this previous year gate question where it shows an beautiful application of Fibonacci series .
So for bit string of length $n$ , the number of string which not contain two consecutive 0’s are there follow Fibonacci series where$ f1 =2 ,f2=3,f3=5 …..fn=fn-1+fn-2$
GATE CSE 2016 Set 1 | Question: 2 - GATE Overflow for GATE CSE
So we can think think this problem as , given first and last bit as 1 for a string of length $10$ we are remaining with $8$ bit .
so the problem reduce to “the number of 8 bit string that do not contain two consecutive 0’s” which is $f8$.
$f4=f3+f2=5+3=8$
$f5=f4+f3=8+5=13$
$f6=f5+f4=13+8=21$
$f7=f5+f6=13+21=34$
$f8=f7+f6=34+21=55$
So our required number of string is $55$.