Discrete Math

From a set A with numbers 1 through 6, how many strictly increasing sequences of three objects can be formed?

From a set A with numbers 1 through 6, how many strictly decreasing sequences of three objects can be formed?

Consider the word formed by the letters in 'memory'. How many unique arrangements of these letters are possible?

Consider the word 'memory'. How many arrangements contain the word 'I'?

From the letters in 'memory', how many arrangements contain both the word 'RAM' and 'I'?

Given 12 books, how can they be distributed among 4 children?

Last year, 26 members of a certain club traveled to England, 26 members traveled to France, and 32 members traveled to Italy. No members traveled to both England and France, 6 members traveled to both England and Italy, and 11 members traveled to both France and Italy. How many members of the club traveled to at least one of these three countries last year?

In a group of 15 people, 12 voted for pizza, 10 for sushi, and 7 for tacos. If some voted for two or three types, calculate the number of people who voted for each combination (voted once, twice, or all three).

Is it possible for 5 people to each be friends with exactly 2 other people?

Is it possible for 5 people to each be friends with exactly 3 other people?

Use the Depth First Search algorithm to find the spanning tree of a graph starting from an arbitrary vertex and traversing through the vertices by following alphabetical order, then backtracking when no more edges are available.

Using a depth-first search algorithm, create a spanning tree for a given graph starting from a specified root vertex.

If we take dominoes, each of which can cover up two of these squares, can we perfectly tile this chessboard after removing two squares from opposite corners?

If you pick any five points on Earth, can you find some closed hemisphere that includes at least four of those five points?

Prove that there is no way to have a lossless compression algorithm that always produces a smaller output for any given input.

For a military mission, engineers are creating an expensive system reliant on a fragile component with a failure probability of 0.32. With 6 installed components working independently in parallel, what is the probability that exactly one of these components works?

A certain union chapter has 32 male and 26 female members, totaling 58 members. If a committee of 7 people is randomly selected, what is the probability that the committee has exactly 6 women?

Suppose a beverage manufacturer adds gold flakes into a beverage at a rate of 800 flakes per litre. Given a 10 millilitre sample taken from a well-shaken 1 litre bottle, what is the probability that the sample contains exactly 2 flakes of gold?

A European roulette wheel has 18 red slots, 18 black slots, and 1 green slot. What is the probability that the first time the ball lands in a red slot occurs on the 5th spin?

Sarah is deciding which courses she wants to take in her next college semester. The probability that she enrolls in an algebra course is 0.3 and the probability that she enrolls in a biology course is 0.7. The probability that she will enroll in an algebra course given that she enrolls in a biology course is 0.4.

Part A: What is the probability that she will enroll in both an algebra course and a biology course?