Exploring Mathematics and Data Science with Technology

(a)

Suppose for a fancy dinner you are given four options as an appetizer, six options as an entree, three options as a beverage, and two options as a dessert. How many different dinners are possible?

(b)

Suppose you could choose an appetizer or a dessert, but not both. How many different dinners are possible in that scenario?

(c)

What if instead you consider the choice to skip any part of the meal, holding the appetizer, entree, beverage, and/or dessert. How many dinners are possible in that case?

(d)

What is the size of the sample space for the experiment of rolling a D6, then flipping a coin, then drawing a card from a shuffled 52-card deck?

(a)

How many permutations are there for the letters \(W,X,Y,Z\text{?}\)

(b)

A list of permutations of the letters \(A,B,C\) may be obtained in SageMath (and Python) by writing import itertools and abcs = ["".join(p) for p in itertools.permutations(["A", "B", "C"])]. Then len(abcs) would equal 6.

Create a list of all strings that are formed by the letters in ["w","x","y","z"]. Print out these strings, and count them using len() to confirm your calculation in the previous task.

(c)

Expressions such as \(5\times 4\times 3\times 2\times 1\) are commonplace in mathematics, so we use the factorial operation as a shorthand:

In SageMath, factorial(5) may be used to compute \(5\times 4\times 3\times 2\times 1=120\text{.}\)

Use a Code cell to count how many permutations are there for the letters of the alphabet \(A\) to \(Z\text{.}\)

(d)

Attempting to check this by computer by actually listing out every permutation as you did for ["w","x","y","z"] would be difficult. (Try it!) What causes this difficulty? What does that suggest about the relationship between mathematics and computer science?

(a)

How many five-character passwords use only uppercase letters?

(b)

How many five-character passwords use only alphanumeric characters? That is, uppercase letters, lowercase letters, or digits, where htr12 is considered different than the password HtR12.

(c)

How many five-character passwords use only alphanumeric characters, when repeated characters are not allowed? (That is, both aabcd and abacd are both not allowed.)

(d)

How many five-character passwords use only alphanumeric characters, when consecutive repeated characters are not allowed? (That is, aabcd is not allowed, but abacd is okay.)

(e)

Consider the experiment where a five-character password is generated using alphanumeric characters (with repeated characters allowed). What's the probability that such a password would only have uppercase letters? Use the n() tool to obtain a decimal approximation.

(f)

The code in 6.2.1 simulates generating 100 thousand passwords and checking if the password contains all uppercase letters. Run the code as-is to confirm that it illustrates what you found in the previous task. Then annotate the code with # comments that explain how it works.

# Add comments like this throughout that explain how this code works.
points = []
all_upper_count = 0
for trial in range(1,100001):
    all_upper = True
    for _ in range(5):
        next_character = randrange(62)
        if next_character >= 26:
            all_upper = False
    if all_upper:
        all_upper_count += 1
    ex_prob = all_upper_count/trial
    points.append( (trial,ex_prob) )

show(
    line(points, color="blue")+line([(1,0.012969),(100000,0.012969)], color="red"),
    axes_labels=["Trials","Exp. Prob."],
)