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This document contains several combinatorics and probability problems. Let's address each of the problems one by one:

### Aufgabe 1:
A three-course meal consists of an appetizer, a main course, and a dessert. You can choose from:
- 3 appetizers,
- 4 main courses,
- 2 desserts (including ice cream).

**a) Total number of possible meal combinations:**
- To find the number of combinations, multiply the number of choices for each course:  
\[
3 \times 4 \times 2 = 24
\]
So, there are 24 possible meal combinations.

**b) Miriam insists on having ice cream as her dessert:**
- Since there are 2 desserts, and one of them is ice cream, there is only 1 choice for dessert. The choices for appetizer and main course remain the same:
\[
3 \times 4 \times 1 = 12
\]
So, under this condition, 12 meal combinations include ice cream as the dessert.

### Aufgabe 2:
EC bank cards have a four-digit PIN code, with digits ranging from 0 to 9. Each digit can be used in any position.

**a) Number of possible PIN combinations:**
- Since there are 10 possible digits for each of the 4 positions, the total number of different PINs is:
\[
10 \times 10 \times 10 \times 10 = 10^4 = 10,000
\]
So, there are 10,000 possible PIN combinations.

**b) Explanation for card withdrawal after three wrong attempts:**
- This is a security measure. After three incorrect attempts, the card is withdrawn to prevent unauthorized access by brute force attempts to guess the PIN.

### Aufgabe 3:
There are 8 ways to travel from point \( P \) to \( Q \), 5 ways from \( Q \) to \( R \), and 6 ways from \( R \) to \( S \).

**Determine the number of different routes from \( P \) via \( Q \) and \( R \) to \( S \):**
- The total number of routes is the product of the number of ways to travel each leg of the journey:
\[
8 \times 5 \times 6 = 240
\]
So, there are 240 different routes from \( P \) to \( S \).

### Aufgabe 4:
There are 13 football matches, and for each match, the outcome can be:
- 1: Home team wins,
- 0: Draw,
- 2: Away team wins.

**a) Number of possible ways to fill out a betting slip:**
- Since there are 3 possible outcomes for each match, the total number of ways to fill out the betting slip is:
\[
3^{13} = 1,594,323
\]
So, there are 1,594,323 possible ways to fill out the betting slip.

### Aufgabe 5:
**a) Constructing a tower with white and red blocks:**
- If you are selecting 4 blocks (stones) and you can choose from red and white, each position in the tower can either be red or white.
- The number of possible towers is \( 2^4 = 16 \) because each stone has 2 possible choices.

**b) Ski race medals:**
- There are 4 skiers competing for the gold, silver, and bronze medals. This is a permutation problem, where the order of medals matters. The number of possible ways to distribute the medals is:
\[
P(4, 3) = 4 \times 3 \times 2 = 24
\]
So, there are 24 ways to distribute the medals among the 4 skiers.

---

Let me know if you'd like further details or explanations for any part of the solution! Here are 5 additional questions to think about:

1. How would the number of combinations in **Aufgabe 1** change if an additional dessert were added?
2. What is the probability of guessing the correct PIN on the first try in **Aufgabe 2**?
3. If in **Aufgabe 3** one route is blocked, how does the number of possible routes change?
4. How would the number of betting slip combinations in **Aufgabe 4** change if one more football match were added?
5. If one more skier entered the competition in **Aufgabe 5b**, how would that affect the number of possible medal distributions?

**Tip:** For probability-related questions, always consider how the total number of outcomes influences the likelihood of a single event happening.

A series of math problems related to combinations and probabilities, including selecting meals, PIN codes, routes, football match predictions, tower construction with blocks, and ski race medal distributions.

Explore combinatorics through problems like meal combinations, PIN code generation, route calculations, football match predictions, and ski race medal distributions. Ideal for high school students (Grades 9-12) learning about probability and counting techniques.

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