GATE CS 2014 Set 1 — Question 59
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Engineering Mathematics → Sets & Combinatorics → Permutations & Combinations
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Question
A pennant is a sequence of numbers, each number being 1 or 2. An n-pennant is a sequence of numbers with sum equal to n. For example, (1,1,2) is a 4-pennant. The set of all possible 1-pennants is {(1)}, the set of all possible 2-pennants is {(2), (1,1)} and the set of all possible 3-pennants is {(2,1), (1,1,1), (1,2)}. Note that the pennant (1,2) is not the same as the pennant (2,1). The number of 10-pennants is ______________.
Correct answer
88.9 to 89.1
Solution
Let be the number of n-pennants.
An n-pennant is a sequence of 1s and 2s that sums to .
The last term in the sequence can be either 1 or 2.Case 1: The last term is 1.
The sum of the preceding terms must be . The number of such sequences is .Case 2: The last term is 2.
The sum of the preceding terms must be . The number of such sequences is .Thus, the recurrence relation is:
Base cases:
(Sequence: (1))
(Sequences: (1,1), (2))Calculating for :
The number of 10-pennants is 89.
An n-pennant is a sequence of 1s and 2s that sums to .
The last term in the sequence can be either 1 or 2.Case 1: The last term is 1.
The sum of the preceding terms must be . The number of such sequences is .Case 2: The last term is 2.
The sum of the preceding terms must be . The number of such sequences is .Thus, the recurrence relation is:
Base cases:
(Sequence: (1))
(Sequences: (1,1), (2))Calculating for :
The number of 10-pennants is 89.
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