GATE CS 2016 Set 1 — Question 12
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Engineering Mathematics → Sets & Combinatorics → Recurrence Relations
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Question
Let be the number of -bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for ?
Correct answer
(B) aₙ = aₙ₋₁ + aₙ₋₂
Solution
Let be the number of valid -bit strings without consecutive 1s.A valid string of length can end in either '0' or '1'.
1.Ends in '0': The preceding bits can be any valid string of length . The number of such strings is .
2.Ends in '1': If the string ends in '1', the bit before it must be '0' (to avoid consecutive 1s). Thus, the string ends in '01'. The preceding bits can be any valid string of length . The number of such strings is .
Total number of strings .This is the standard recurrence for Fibonacci numbers (with different initial conditions depending on definition).Continue learning with Success Tracker
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