We have 16 cards, 4 each of red, green, yellow and blue. How many choices we have to selecting
3 out these 16 cards, satisfying that these three cards are different colors and red appears at
most once.


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.

---

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.

We have 16 cards, 4 each of red, green, yellow, and blue. How many choices do we have for selecting 3 out of these 16 cards, satisfying that these three cards are different colors and red appears at most once?

Explore how to solve a combinatorial problem involving the selection of 3 cards with different colors from a set of 16 cards, with specific conditions on the appearance of red cards. The solution uses fundamental counting principles and combination formulas, suitable for Grades 10-12.

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