Discrete Math

Using generating functions, determine how many ways there are to combine 10 candies when the candies are red, blue, and green with the conditions: even number of red candies, more than six blue candies, and less than three green candies.

How many ways can we pick n shirts out of four different shirts? Provide answers for n = 0, 1, 2, 3, 4, 5.

How many ways can we pick n identical socks out of five identical socks? Provide answers for n = 0, 1, 2, 3, 4, 5, 6.

Find the number of non-negative solutions to $x_1 + x_2 + x_3 = 6$, where $x_1 \leq 4$, $x_2 \leq 3$, and $x_3 \leq 5$ using generating functions.

Explain how to represent a graph using an adjacency list, adjacency matrix, and incidence matrix in JavaScript.

Represent a graph using an adjacency matrix given a graph with 5 vertices.

Represent a graph using an adjacency list for a given graph with 5 vertices.

A fairly standard problem that you're likely to encounter all the time revolves around connectivity.

What sort of algorithms can be used to allow a computer to efficiently solve the problem of determining connectivity between two vertices?

What is the path of the least length between two vertices in a graph (shortest path problem)?

Given a set of colors, can we assign a color to each vertex such that no two neighbors are assigned the same color (vertex coloring problem)?

Does a path exist that uses every edge exactly once in a graph, and what algorithms exist to determine this?

What about the existence of a path that uses every vertex exactly once in a graph, and why is this problem characterized by exponential time algorithms?

Explain the difference and use cases for undirected and directed graphs with examples such as social networks and street maps.

Describe how adjacency matrices and adjacency lists are used to represent graphs and compare their time and space complexities.

In 2005, there were a thousand rabbits on an island. The population grows 8% every year. At this rate, how many rabbits will there be on the island by 2020?

The value of a new car in 2015 was $40,000. It depreciates 7% each year. How much will the car be worth in 2024?

John bought a new home in 2002. The value of the home increases 4% each year. If the price of the house is $225,000 in 2015, how much did he pay for it in 2002?

A sample contains a certain amount of bacteria. The bacteria doubles every 20 minutes. At this rate, how many counts of bacteria will there be in 3 hours?

A sample contains 100 counts of bacteria. The bacteria triples every 15 minutes. How much bacteria will there be in 1 hour?