Discrete Math

Find three different spanning trees for a graph with five vertices, seven edges, and multiple overlapping circuits by removing three edges.

A company requires reliable internet and phone connectivity between their five offices. They decide to lease dedicated lines from the phone company. The phone company will charge for each link made. The cost in thousands of dollars per year are shown below in the graph. Find a spanning tree for this graph that ensures connectivity between all offices without forming any circuits.

Prove the function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x) = \pi x - e$ is a bijection and find the inverse.

Prove that a function $f : [0, 1] \to [2, 4]$ is a bijection by showing it is well-defined, injective, and surjective.

Find the generating function for a sequence given recursively by: $a_n = 2a_{n-1} + 4a_{n-2}$ with initial terms $a_0 = 1$, $a_1 = 3$, and $a_2 = 10$.

Find the generating function for a sequence given recursively by: $a_n = a_{n-1} + 2a_{n-2} + 3$ with initial terms $a_0 = 2$ and $a_1 = 2$.

Solve the recurrence relation $a_n = 3a_{n-1} - 2a_{n-2}$ with initial conditions $a_0 = 1$ and $a_1 = 3$.

Compute $(3x + 2y)^5$ using Pascal's Triangle.

Expand $(x + 5)^5$ using Pascal's Triangle.

Prove that if $x$ is odd, then $x^2$ is odd.

Prove that if $x$ and $y$ are odd, then $xy$ is odd.

Prove: If $n$ is an odd integer, then $n^2$ is odd.

Prove: The sum of two even integers is even.

How many permutations of the 10 digits (0 through 9) have at least one of the patterns 60, 04, or 42 appear consecutively?

We have 150 students who drink three beer brands: A, B, and C. Given: 58 students drink brand A, 49 drink brand B, 57 drink brand C, 14 drink both A and C, 13 drink both A and B, 17 drink both B and C, and 4 drink all three brands A, B, and C. Determine how many students drink none of these brands.

A bag consists of 8 red marbles, 7 blue marbles, 6 green marbles, and 4 yellow marbles. What is the probability of selecting a red marble?

What is the probability of selecting a blue marble on the first try and then a green marble on the second try with replacement?

What is the probability of selecting a yellow marble on the first try and then a red marble on the second try without replacement?

What is the probability of selecting two blue marbles with replacement?

What is the probability of selecting two green marbles without replacement?