C960 Discrete Mathematics II

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Free C960 Discrete Mathematics II Questions

How many handshakes occur when 12 people each shake hands with every other person exactly once?

Explanation

When each person shakes hands with every other person exactly once, the total number of handshakes can be calculated using the combination formula $(\begin{matrix} n \\ 2 \end{matrix})$ , which counts the number of ways to choose 2 people out of nnn to shake hands. For 12 people, this is:

$(\begin{matrix} 12 \\ 2 \end{matrix}) = \frac{12.11}{2} = 66$

Solve the recurrence aₙ = 4aₙ₋₁ − 4aₙ₋₂ with a₀=1, a₁=4

$(n + 1) 2^{n}$
$2^{n}$
$n 2^{n}$
$(2 n + 1) 2^{n - 1}$

Explanation

Find $X$ if $7^{x} \equiv 2 (m o d 13)$ (discrete logarithm problem).

Explanation

How many Hamiltonian cycles are there in the complete graph K₅?

Explanation

A Hamiltonian cycle in a graph visits each vertex exactly once and returns to the starting vertex. In a complete graph $k_{n}$ , every vertex is connected to every other vertex.

What is the probability that a random 5-letter string over the alphabet $\{A, B, C\}$ contains at least two consecutive A’s?

79/243
8/27
1/3
2/5

Explanation

Using Fermat’s Little Theorem, compute $3^{100}$ mod 7.

Explanation

Fermat’s Little Theorem states that if ppp is a prime number and aaa is an integer not divisible by p, then $a^{p - 1}$ −1≡1(mod p).

How many edges are in a 5-regular graph with 12 vertices?

Explanation

In a k-regular graph with nnn vertices, each vertex has degree $k$ . The sum of all vertex degrees is:

How many 5-card poker hands contain exactly three of a kind (but not a full house)?

54,912
3744
123,552
2,598,960

Explanation

Exactly three of a kind means the hand contains three cards of one rank and two other cards of different ranks (and those two must not form a pair, otherwise it would be a full house). Count by choices: pick the rank for the triple $(\begin{matrix} 13 \\ 1 \end{matrix}) = 13$ , pick which 3 of the 4 suits for that rank $((\begin{matrix} 4 \\ 3 \end{matrix}) = 4)$ , choose 2 distinct ranks from the remaining 12 ranks for the other cards $((\begin{matrix} 12 \\ 2 \end{matrix}) = 66)$ , and for each of those ranks choose 1 of the 4 suits $4^{2} = 16$ . Multiply: 13×4×66×16=54,912. (For reference, the total number of 5-card hands is 2,598,960, which shows this count is a fraction of all possible hands.)

What is the chromatic polynomial of the complete bipartite graph K_{3,3}?

$x (x - 1) {(x - 2)}^{4}$
$x (x - 1) (x - 2) {(x - 3)}^{3}$
$x (x - 1) (x - 2) (x - 3) {(x - 4)}^{2}$
$x {(x - 1)}^{2} {(x - 2)}^{3}$

Explanation

10.

What is the value of the partition function p(8) (number of ways to write 8 as a sum of positive integers, disregarding order)?

Explanation

The partition function p(n)p(n)p(n) counts the number of ways to write nnn as a sum of positive integers, ignoring the order of the summands. For n=8n = 8n=8, the partitions are:

1.8

2.7 + 1

3.6 + 2

4.6 + 1 + 1

5.5 + 3

6.5 + 2 + 1

7.5 + 1 + 1 + 1

8.4 + 4

9.4 + 3 + 1

10.4 + 2 + 2

11.4 + 2 + 1 + 1

12.4 + 1 + 1 + 1 + 1

13.3 + 3 + 2

14.3 + 3 + 1 + 1

15.3 + 2 + 2 + 1

16.3 + 2 + 1 + 1 + 1

17.3 + 1 + 1 + 1 + 1 + 1

18.2 + 2 + 2 + 2

19.2 + 2 + 2 + 1 + 1

20.2 + 2 + 1 + 1 + 1 + 1

21.2 + 1 + 1 + 1 + 1 + 1 + 1

22.1 + 1 + 1 + 1 + 1 + 1 + 1 + 1

Counting carefully, there are 22 distinct partitions of 8.

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