How would I formalize this combinatorics question?

Question

We have workouts, durations, and difficulties and we want to know how many unique combinations of these items we can form.

Workout	Duration	Difficulty
Strength	10mins	Beginner
Yoga	15mins	Intermediate
Stretching	20mins	Advanced
Biking
Running
Rowing

There are three different ways these options can be combined.

1. The user picks three things. The user picks one option for each of the three given categories: $6 \times 3 \times 3 = 54$
2. The user picks two things. The user selects options for 2 of the three areas, either workout and duration ($6 \times 3 = 18$) or workout and difficulty ($6 \times 3 = 18$) or duration and difficulty ($3 \times 3 = 9$)
   1. Each combination generates a distinct result i.e. selecting 10mins and strength would give the user a different result than selecting 10mins, strength, and beginner.
3. The user picks one thing. The user selects one option for the workout category ($6$) or one option for the duration ($3$) or one option for the difficulty ($3$)

This gives us:

$$ 54+(18+18+9)+(6+3+3) = 111 $$

I'm comfortable with this casing, but there's definitely a formalization of this problem that I'm missing.

Answer 1

Workout	Duration	Difficulty
Strength	10mins	Beginner
Yoga	15mins	Intermediate
Stretching	20mins	Advanced
Biking	NOTHING	NOTHING
Running
Rowing
NOTHING

Add an additional option to each column. Then the user picks one thing from each column, one of which may be NOTHING. The number of ways to do that is \begin{align} & (6+1)(3+1)(3+1) \\[8pt] = {} & (6\times3\times3) \\ & {}\qquad{} + (6\times3\times1) + (6\times1\times3) + (1\times3\times3) \\ & {}\qquad{} + (6\times1\times1) + (1 \times3\times1) + (1\times1\times3) \\ & {}\qquad{} + (1\times1\times1). \end{align} Then if you want to exclude the possibility of choosing NOTHING all three times, subtract $1.$

Algebra tells us that \begin{align} & (a+1)(b+1)(c+1) \\[8pt] = {} & abc + ab + ac + bc + a + b + c + 1. \end{align}

I don't know whether this to some extent begins to answer your question or not.