C960 Discrete Mathematics II
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Free C960 Discrete Mathematics II Questions
What is the expected number of coin flips until you get two heads in a row?
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4
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6
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8
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10
Explanation
To find the expected number of flips to get two consecutive heads, we can use a state-based approach. Let be the expected flips starting from no heads, and the expected flips starting with one head.
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 ways can 7 people sit around a circular table if two specific people must sit together?
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240
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360
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480
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720
Explanation
For circular arrangements, we usually fix one seat to account for rotations, giving (n−1)! arrangements for nnn people.
Here, there are 7 people with two specific people required to sit together. Treat these two as a single "block," reducing the problem to 6 objects (the block + 5 other people). The number of circular arrangements is:
Which of the following languages requires a linear bounded automaton (LBA)?
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The set of palindromes over {0,1}
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The set of strings of the form
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The set of all binary strings
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The set of all strings with an even number of 0s
Explanation
A linear bounded automaton (LBA) is a type of Turing machine whose tape usage is limited to a linear function of the input length. LBAs recognize context-sensitive languages, which are strictly more powerful than context-free languages but less powerful than general Turing machines.
Palindromes over {0,1} are context-free, so a pushdown automaton can handle them.
Strings of the form are context-sensitive but not context-free, so they require an LBA.
What is the coefficient of x⁸ in the expansion of (x + 2 + x⁻¹)⁸?
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1
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70
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56
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64
Explanation
Using Fermat’s Little Theorem, compute mod 7.
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1
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2
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3
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4
Explanation
Fermat’s Little Theorem states that if ppp is a prime number and aaa is an integer not divisible by p, then −1≡1(mod p).
How many 5-digit numbers have digits that sum to 20?
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106
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121
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126
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210
Explanation
In a group of 30 people, what is the minimum number that guarantees at least 4 people share the same birth month?
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37
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10
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13
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4
Explanation
This is an application of the Pigeonhole Principle. There are 12 months (pigeonholes) and 30 people (pigeons). To guarantee at least 4 people in the same month, consider the worst-case scenario where no month has 4 people: each month could have at most 3 people.
12 months×3 people per month=36
Wait, we need the minimum number that guarantees 4 people in a month. In the worst case, if each month has at most 3 people, then 12 × 3 = 36 people could avoid having 4 in a month. But we only have 30 people, which is less than 36.
To guarantee 4 people in a month, the minimum number of people needed is 3 × 12 + 1 = 37.
Since we have only 30 people, we cannot guarantee 4 in a month.
If the question is asking more generally: the formula is:
Minimum number to guarantee k in a box=(k−1)⋅n+1
Here, k=4, n=12:
(4−1)⋅12+1=37
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.
How many ways can you arrange the letters in “MISSISSIPPI”?
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34,650
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36,300
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37,800
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40,000
Explanation
The word MISSISSIPPI has 11 letters in total. The counts of each letter are:
M = 1
I = 4
S = 4
P = 2
The total number of distinct arrangements is given by dividing the factorial of the total letters by the factorial of the frequency of each repeated letter:
Number of arrangements=
so,
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