Question

Let L = {a, bb}
How many strings of length 10 are present in L* ?

Answer 1

Length 1: only one string - "a"

Length 2: "aa", "bb" - 2 strings

Length 3: "aaa", "bba", "abb"

Length (n) = Length (n-1) + Length(n-2),

as we get a string of length $n$, by appending "a" to a string of length $n-1$ as well as by appending "bb" to a string of length $n-2$. (This works only because 'a' and 'bb' doesn't have a common character). 

So, 

n	No. of strings
1	1
2	2
3	3
4	5
5	8
6	13
7	21
8	34
9	55
10	89

We get no. of strings = 89 (10th Fibonacci number) 

Comments

@radha there are mistakes 

as instead of 10C8 it should be 9C8, similarly 10C6 should be 8C6

and you left one option for 0 bb

check it, https://gateoverflow.in/31927/tifr-2015-maths-b-10

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Sir I got the the logic for first case since we have bb together so 9 choices but second logic I couldn't get since I have already considered the case where all 10 positions are a's so only 1 way .

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those strings will have 2 bb will never be same as those having 1 bb. there will be no relation with previous.