To find the number of trailing zeroes in the product $12 \times 23 \times 34 \times \dots \times 2829$, we need to focus on how many factors of 10 are present in the product. Each factor of 10 is the result of multiplying a 2 and a 5.

To calculate the number of trailing zeroes in this expression, we need to count how many multiples of 5 are in the given product range.

Step-by-step approach:

1. Numbers divisible by 5: We need to find how many multiples of 5 are there in the sequence $12, 13, 14, ..., 2829$. For this, we divide the largest number (2829) by 5: $\frac{2829}{5} = 565.8 \Rightarrow 565 \text{ multiples of 5}.$ So there are 565 multiples of 5.

2. Numbers divisible by $5^2$ (25): Some numbers contribute more than one factor of 5, such as multiples of $5^2 = 25$. We divide by 25: $\frac{2829}{25} = 113.16 \Rightarrow 113 \text{ multiples of 25}.$

3. Numbers divisible by $5^3$ (125): We continue by counting the multiples of $5^3 = 125$: $\frac{2829}{125} = 22.632 \Rightarrow 22 \text{ multiples of 125}.$

4. Numbers divisible by $5^4$ (625): Multiples of $5^4 = 625$ contribute additional factors of 5: $\frac{2829}{625} = 4.526 \Rightarrow 4 \text{ multiples of 625}.$

5. Numbers divisible by $5^5$ (3125): Since $3125 > 2829$, there are no multiples of $3125$.

Total factors of 5:

The total number of factors of 5 is the sum of the above: $565 + 113 + 22 + 4 = 704.$

Thus, there are 704 factors of 5 in the product $12 \times 23 \times \dots \times 2829$.

Number of trailing zeroes:

Since each trailing zero is produced by a pair of factors of 2 and 5, and there are always more factors of 2 than factors of 5, the number of trailing zeroes is the same as the number of factors of 5. Therefore, the number of trailing zeroes is 704.

However, there seems to be a mismatch between the given options and the actual number calculated. Based on the question and the possible options (which may suggest a different method or focus), I may have misunderstood the problem's intended scope. Let me reanalyze the focus or confirm the method.