To solve the problem, let's break it down step by step:

Step 1: Understand the Problem

You have 16 cards, with 4 each of red, green, yellow, and blue. You need to select 3 cards such that:

1. The three cards are of different colors.
2. The red card appears at most once.

Step 2: Consider Possible Cases

There are two possible cases to consider:

1. Case 1: No red card is selected.
2. Case 2: Exactly one red card is selected.

Step 3: Count the Number of Choices for Each Case

Case 1: No red card is selected

• Since we need 3 cards of different colors, we can choose from the 3 remaining colors (green, yellow, blue).
• The number of ways to choose 3 colors out of these 3 is exactly 1 (since all colors must be chosen).
• From each of these 3 colors, we can pick any of the 4 available cards.
• Therefore, the number of combinations in this case is: $4 \times 4 \times 4 = 4^3 = 64$

Case 2: Exactly one red card is selected

• We need to choose 2 other colors from the remaining 3 (green, yellow, blue).
• The number of ways to choose 2 colors out of these 3 is $\binom{3}{2}$.
• From each of these 2 chosen colors, we can pick any of the 4 available cards.
• The number of combinations in this case is: $\binom{3}{2} \times 4 \times 4 \times 4 = 3 \times 4 \times 4 = 48$

Step 4: Add the Results from Both Cases

The total number of choices is the sum of the cases: $64 \text{ (Case 1)} + 48 \text{ (Case 2)} = 112$

Final Answer:

There are 112 different ways to select 3 cards from the 16, satisfying that the three cards are different colors and red appears at most once.

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Do you want more details or have any questions? Here are some related questions you might find interesting:

1. What if we wanted exactly two red cards?
2. How many ways can we select 3 cards if no restrictions on colors are applied?
3. How would the answer change if we had 5 cards of each color?
4. What is the probability of drawing exactly one red card when choosing 3 cards at random?
5. How many ways can we arrange 3 cards of different colors in a line?

Tip: When solving problems involving combinations and restrictions, breaking the problem into cases often simplifies the solution.