Fundamental Counting Principle

A logistics firm generates a 4-character tracking code, $c_1 c_2 c_3 c_4$, for express parcels according to the following rules:
- The first character, $c_1$, must be a non-zero digit ($1$ through $9$).
- The second character, $c_2$, must be a vowel chosen from the set $\{\text{A, E, I, O, U}\}$.
- The third character, $c_3$, must be an even digit ($0, 2, 4, 6, 8$).
- The fourth character, $c_4$, must be a consonant chosen from the set $\{\text{B, F, G, K, M, P}\}$.

If no digit may be repeated within the code, how many different tracking codes can be created?

A diner offers a three-course meal deal where a customer selects $1$ appetizer from a choice of $4$, $1$ main dish from a choice of $5$, and $1$ dessert from a choice of $3$. How many different three-course meal combinations can a customer order?

A security system requires each employee to create a 6-character access code consisting of 3 non-zero digits followed by 3 letters from the standard 26-letter English alphabet. The access code must satisfy the following restrictions:

1. The first digit must be odd.
2. No two adjacent digits in the code can be identical.
3. The first letter must be a vowel ($A, E, I, O, U$), the second letter must be a consonant, and the third letter cannot be identical to the first letter.

How many distinct access codes can be created under these rules?

A security analyst is designing a 4-character access code consisting of a digit, followed by two letters, followed by another digit. The code must be constructed according to the following rules:

- The first character must be a prime digit chosen from the set of single-digit integers $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.
- The second and third characters must be distinct uppercase letters selected from the set $\{A, B, C, D, E\}$.
- The fourth character must be an odd digit chosen from the set of single-digit integers $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$, such that it is not equal to the first digit.

How many different 4-character access codes can be created following these rules?

An airline creates a $5$-character identification code for each of its cargo routes. The characters in the code must satisfy the following sequential rules from left to right:

1. The first character must be a non-zero even digit.
2. The second and third characters must be distinct uppercase vowels chosen from the set $\{A, E, I, O, U\}$.
3. The fourth character must be a single-digit prime number.
4. The fifth character must be a single digit ($0$ through $9$) such that the sum of the first digit and the fifth digit is odd.

How many such unique identification codes can the airline create?

A tech company generates a 5-digit security badge number, represented as $d_1 d_2 d_3 d_4 d_5$, using digits from $0$ to $9$, inclusive. The badge numbers must satisfy the following constraints:
- The first digit, $d_1$, cannot be $0$ or $1$.
- The second and third digits, $d_2$ and $d_3$, must be distinct even digits.
- The fourth digit, $d_4$, must be an odd digit strictly greater than $3$.
- The fifth digit, $d_5$, can be any digit except that it cannot be equal to $d_1$.

How many different 5-digit badge numbers can be created under these rules?

A medical research laboratory creates a 4-digit identification code, $d_1 d_2 d_3 d_4$, for sample trays using digits selected from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$. The digits in each code must satisfy the following conditions:
- The first digit, $d_1$, must be an odd prime number.
- The second digit, $d_2$, must be an even integer.
- All four digits in the code must be distinct.

How many different identification codes can be created under these rules?

A furniture manufacturer allows customers to customize a dining set by selecting options from four categories:
- Tabletop shape: rectangular, oval, or round (3 choices)
- Wood finish: oak, walnut, cherry, or maple (4 choices)
- Leg design: tapered, hairpin, or turned (3 choices)
- Number of chairs: 4, 6, or 8 (3 choices)

However, due to space constraints, a round tabletop cannot be paired with a set of 8 chairs. How many distinct dining set configurations can a customer assemble?

An artisan tea shop allows customers to order a custom tea blend prepared through three sequential choices:

1. Base Tea: Choose $1$ of $4$ available tea types (Black, Green, Oolong, or White).
2. Flavor Infusion: Choose $1$ of $5$ available herbal flavors. However, if Black tea is chosen as the base, only $3$ of these $5$ herbal flavors can be selected.
3. Sweetener: Choose $1$ of $3$ natural sweeteners (Honey, Stevia, or Maple Syrup), or choose to have no sweetener.

How many different custom tea blends can a customer create?

An architectural firm offers custom townhouse design packages created by making sequential choices across three stages: Exterior Facade Material (4 choices), Floor Plan Layout (5 choices), and Interior Lighting Scheme (6 choices).

However, two design constraints apply:
1. If the first exterior facade material option is chosen, the fifth floor plan layout option cannot be selected.
2. The sixth interior lighting scheme option can only be selected if the fifth floor plan layout option is chosen.

How many different valid townhouse design packages can a client create?