The word BIRD is written seven times in a row. How many times does the (not necessarily contiguous) sub-sequence BIRD appear?

To find how many times the subsequence "BIRD" appears in the sequence "BIRDBIRDBIRDBIRDBIRDBIRDBIRD", we can use a combinatorial approach.

### Step 1: Understanding the problem
Given a sequence "BIRDBIRDBIRDBIRDBIRDBIRDBIRD", which consists of "BIRD" written seven times, we need to count the number of distinct ways to choose one "B", one "I", one "R", and one "D" from the sequence such that these letters appear in that order.

### Step 2: Counting occurrences of each letter
In the sequence, each letter appears 7 times:
- There are 7 "B"s
- There are 7 "I"s
- There are 7 "R"s
- There are 7 "D"s

### Step 3: Combinatorial approach
For each "B", we need to select one "I" that appears after it, one "R" that appears after the chosen "I", and one "D" that appears after the chosen "R". 

1. **Choosing "B":** There are 7 choices.
2. **Choosing "I":** For each "B", there are \(7\) positions left for "I" (but the actual count will decrease as we move further right).
3. **Choosing "R":** For each chosen "I", there are \(7\) positions left for "R".
4. **Choosing "D":** For each chosen "R", there are \(7\) positions left for "D".

The general formula for counting the number of ways to select letters in order is:

\[
\text{Total subsequences} = \sum_{1 \leq i_1 < i_2 < i_3 < i_4 \leq 28} 1
\]

where \( i_1, i_2, i_3, i_4 \) represent the positions of "B", "I", "R", "D" respectively.

### Step 4: Calculating the result
Since each choice of "B" gives 7 choices for "I", 7 choices for "R", and 7 choices for "D", the number of subsequences is:

\[
7 \times 6 \times 5 \times 4 = 840
\]

However, this result is the wrong approach because we overcounted by not accounting for different positions effectively. Instead, we use a binomial coefficient formula:

### Step 5: Binomial coefficient method
We choose positions for "B", "I", "R", and "D" in a sequence of 28 letters. The correct way to count these is by the combination formula:

\[
\text{Total subsequences} = \binom{28}{4} = \frac{28 \times 27 \times 26 \times 25}{4 \times 3 \times 2 \times 1} = 20475
\]

### Final Answer:
The total number of subsequences "BIRD" in the sequence "BIRDBIRDBIRDBIRDBIRDBIRDBIRD" is **20475**.

---

Do you want details or have any questions? 

Here are 8 related questions you might find interesting:
1. What if "BIRD" was repeated 10 times? How would that change the number of subsequences?
2. How would the count change if the letters of "BIRD" were not distinct?
3. Can you explain the binomial coefficient used in this problem?
4. What would the result be if the word was "AB" repeated several times?
5. How can this concept be applied to DNA sequencing?
6. What if we consider non-consecutive "BIRD"? How would the count change?
7. How does this problem relate to Pascal's Triangle?
8. What if we want to find how many times the subsequence "BIR" appears?

**Tip:** In combinatorics, breaking a problem into steps based on order and constraints helps simplify complex counting problems.

Learn how to count the number of subsequences "BIRD" in a sequence "BIRDBIRDBIRDBIRDBIRDBIRDBIRD" using combinatorial methods. Explore the application of binomial coefficients in combinatorics to solve this problem. Suitable for advanced high school students and enthusiasts of combinatorial mathematics.

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