To find the number of trailing zeros in the product $12 \times 24 \times 36 \times \dots \times 2550$, we need to count the number of times 10 appears in this product. Since each 10 is created by a pair of 2 and 5, and there are always more factors of 2, the number of zeros at the end of the product depends on the number of factors of 5. Steps:

1. Identify terms that contribute factors of 5: We look at multiples of 5 in the sequence and count their factors of 5:

• $5^{10}$ contributes 10 factors of 5
• $10^{20}$ contributes 20 factors of 5
• $15^{30}$ contributes 30 factors of 5
• $20^{40}$ contributes 40 factors of 5
• $25^{50}$ contributes 100 factors of 5

2. Add up the factors of 5: $10 + 20 + 30 + 40 + 100 = 200$

So, the product has $\textbf{200 trailing zeros}$.