C960 Discrete Mathematics II
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Free C960 Discrete Mathematics II Questions
Which of the following is the correct Master Theorem case for T(n) = 9T(n/3) + n²?
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Case 1 → Θ(n³)
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Case 2 → Θ(n³ log n)
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Case 3 → Θ(n²)
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Case 4 → Θ(n² log n)
Explanation
Which of the following problems is NP-complete?
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Sorting a list of numbers
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Finding the shortest path in a graph
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3-SAT (Boolean satisfiability with clauses of size 3)
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Checking if a number is prime
Explanation
A problem is NP-complete if it is in NP (its solutions can be verified in polynomial time) and every problem in NP can be reduced to it in polynomial time. Among the options:
Sorting can be done in polynomial time, so it is in P, not NP-complete.
Shortest path can also be solved in polynomial time using Dijkstra’s or Bellman-Ford algorithms.
3-SAT is the canonical NP-complete problem: verifying a satisfying assignment is polynomial-time, and it is known that any NP problem can be reduced to 3-SAT.
Primality checking is in P (AKS algorithm), so it is not NP-complete.
How many Hamiltonian cycles are there in the complete graph K₅?
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12
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24
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60
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120
Explanation
A Hamiltonian cycle in a graph visits each vertex exactly once and returns to the starting vertex. In a complete graph , every vertex is connected to every other vertex.
In a single-elimination tournament with 16 teams, how many games are played until a champion is determined?
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15
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16
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8
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31
Explanation
In a single-elimination tournament, each game eliminates exactly one team. Starting with 16 teams, we need to eliminate 15 teams to determine a champion. Since each game eliminates one team, the total number of games played is equal to the number of teams eliminated, which is 16−1=15. This formula holds for any single-elimination tournament: total games = total teams − 1.
How many comparisons does Merge Sort make in the worst case for n=8 elements?
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8
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12
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17
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20
Explanation
Explanation:
Merge Sort is a divide-and-conquer algorithm that splits the array into two halves, recursively sorts each half, and then merges the sorted halves.
The worst-case number of comparisons can be computed using the recursive formula for comparisons C(n):
where n−1 comparisons are made during the merge step.
For n=8n
1.Split 8 into two halves of 4:C(8)=C(4)+C(4)+7
2.For C(4): split into 2 + 2:C(4)=C(2)+C(2)+3
3.For C(2): split into 1 + 1:C(2)=C(1)+C(1)+1=0+0+1=1
Now compute step by step:
C(4)=1+1+3=5
C(8)=5+5+7=17
What is the output of this greedy algorithm for the coin change problem (coins: 1, 5, 10, 25) when making 67 cents?
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2 quarters, 1 dime, 1 nickel, 2 pennies
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2 quarters, 1 dime, 1 nickel, 3 pennies
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2 quarters, 1 dime, 1 nickel, 7 pennies
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2 quarters, 1 dime, 1 nickel, 1 penny
Explanation
In the greedy algorithm for the coin change problem, we always choose the largest coin denomination that does not exceed the remaining amount. Starting with 67 cents:
Take 2 quarters (25 × 2 = 50) → remaining 17 cents.
Take 1 dime (10) → remaining 7 cents.
Take 1 nickel (5) → remaining 2 cents.
Take 2 pennies (1 × 2 = 2) → remaining 0 cents.
So the algorithm selects: 2 quarters, 1 dime, 1 nickel, and 2 pennies. This greedy approach gives the optimal result for standard U.S. coin denominations.
Which of the following is a context-free language but NOT regular?
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{ww | w ∈ {0,1}}
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{aⁿbⁿcⁿ | n ≥ 0}
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{wwʳ | w ∈ {a,b}}
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{aⁿbⁿ | n ≥ 0}
Explanation
A context-free language (CFL) is one that can be generated by a context-free grammar, while a regular language can be recognized by a finite automaton. The language consists of equal numbers of ’s followed by equal numbers of ’s. A finite automaton cannot keep track of how many ’s have been seen, but a pushdown automaton (which has a stack) can—by pushing for each aaa and popping for each . Therefore, this language is context-free but not regular. The other languages either have no dependency between symbol counts (making them regular) or require balancing across three symbols (which is not even context-free).
What is the value of the Ramsey number R(4,3)?
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7
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8
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9
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10
Explanation
The Ramsey number is the smallest integer such that any graph on vertices contains either a clique of size mmm or an independent set of size nnn. For , it is known that:
This means that in any graph of 9 vertices, either there is a 4-vertex clique or a 3-vertex independent set. With 8 vertices, it is possible to construct a graph that avoids both, so 9 is the minimal number satisfying this property.
How many ways can you distribute 15 identical chapatis to 6 different students if each student gets at least one chapati?
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5005
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2002
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364
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1365
Explanation
This is a stars and bars problem with the restriction that each student gets at least one chapati. Let be the number of chapatis received by the -th student. We have:
What is the number of onto functions from a set of 10 elements to a set of 3 elements?
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59049
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19683
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270
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27
Explanation
The number of onto functions from a set of size mmm to a set of size nnn can be calculated using inclusion-exclusion:
Number of onto functions=
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