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 After three incorrect attempts, the card is withdrawn to prevent unauthorized access by brute force attempts to guess the PIN.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.