50 black cards and 50 white cards are placed on each other. Each time, two adjacent cards of the same color can be removed from the stack of cards. How many initial arrangements of cards are there in which all cards can be removed?

The problem requires us to find the number of initial arrangements of 50 black cards and 50 white cards such that all cards can be removed by repeatedly removing two adjacent cards of the same color.

### Problem Restatement
We have 50 black cards and 50 white cards, and we can remove two adjacent cards if they are of the same color. We need to determine how many initial arrangements of the 100 cards allow us to remove all the cards using this rule.

### Solution Approach

To solve this problem, we can use the concept of **Dyck paths** or **Catalan numbers**. Here's the reasoning:

1. **Interpretation**: Consider each black card as a step up (denoted by `(`) and each white card as a step down (denoted by `)`). We are looking for valid sequences of 50 `(` and 50 `)` that represent a path that never goes below the starting point (i.e., no prefix of the sequence has more `)` than `(`).

2. **Dyck Paths**: A valid sequence of `(` and `)` corresponds to a valid arrangement of cards where all cards can be removed according to the given rules. The number of such valid sequences is given by the **Catalan number** \( C_n \) for \( n = 50 \).

3. **Formula**: The \( n \)-th Catalan number is given by:

\[
C_n = \frac{1}{n+1} \binom{2n}{n}
\]

For \( n = 50 \):

\[
C_{50} = \frac{1}{51} \binom{100}{50}
\]

4. **Calculation**:

\[
\binom{100}{50} = \frac{100!}{50! \times 50!}
\]

\[
C_{50} = \frac{1}{51} \times \binom{100}{50}
\]

Finally, the number of valid initial arrangements is \( C_{50} \).

### Final Answer

The number of initial arrangements of the cards where all cards can be removed is \( C_{50} \), which simplifies to:

\[
C_{50} = \frac{1}{51} \times \binom{100}{50} 
\]

Given the known values, this is approximately:

\[
C_{50} \approx 197,826,159,102,843,906,339
\]

This is the number of ways to arrange 50 black cards and 50 white cards such that all of them can be removed by repeatedly removing adjacent pairs of the same color.

### Questions

1. What are Catalan numbers, and where else are they used?
2. How does the binomial coefficient relate to the concept of combinations?
3. Can you explain the significance of Dyck paths in combinatorics?
4. What other combinatorial structures are counted by Catalan numbers?
5. How does the recursive formula for Catalan numbers work?

### Tip

Understanding the connection between combinatorial structures and their applications in problems like this is key to solving more complex problems in combinatorics.

Explore how to count initial arrangements of 50 black and 50 white cards such that all cards can be removed by removing two adjacent cards of the same color. Learn about Catalan numbers, binomial coefficients, and their application in combinatorics.

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