GATE | GATE CS 2008 | Question 78

Let xn denote the number of binary strings of length n that contain noconsecutive 0s. Which of the following recurrences does Xn satisfy?

Q79
(A) A
(B) B
(C) C
(D) D

Answer: (D)
Explanation:

  • For n = 1 i.e. binary strings of length 1, strings are ‘0’, ‘1’.
    So, X1 = 2
  • For n = 2 i.e. binary strings of length 2, strings are ’01’ , ’10’ , ’11’.
    (here string ’00’ will be rejected because it has consecutive zeros).
    So, X2 = 3
  • For n = 3 i.e. binary strings of length 3, strings are ‘010’ , ‘011’ , ‘101’ , ‘110’ , ‘111’ .

0 01 (rejected)
0 10
0 11
1 01
1 10
1 11
So, X3 = 5
This shows that X3 = X2 + X1
Therefore, X(n) = X(n – 1) + X(n – 2)
Please comment below if you find anything wrong in the above post.

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