Find the number of zeroes in:
`100^(1)xx99^(2)xx98^(3)xx97^(4)xx….xx1^(100)`

Answer

To find the number of trailing zeros in the expression \(100^1 \times 99^2 \times 98^3 \times \ldots \times 1^{100}\), we need to determine how many times the factors of 10 are present in the product. Since \(10 = 2 \times 5\), we will count the number of factors of 2 and 5, and the minimum of these counts will give us the number of trailing zeros.

### Step-by-Step Solution:

1. **Count the number of factors of 5:**
- For each \(n\) from 1 to 100, we need to find the contribution of \(n\) to the count of factors of 5 in \(n^k\) where \(k\) is the power of \(n\) in the product.
- The contribution of \(n\) to the count of 5s is given by:
\[ \text{Contribution of } n = k \times \left\lfloor \frac{n}{5} \right\rfloor + k \times \left\lfloor \frac{n}{25} \right\rfloor + k \times \left\lfloor \frac{n}{125} \right\rfloor + \ldots \]
- We will calculate this for each \(n\) from 1 to 100.

2. **Calculate contributions:**
- For \(n = 100\), \(k = 1\):
\[ \text{Factors of 5} = 1 \times (20 + 4) = 24 \]
- For \(n = 99\), \(k = 2\):
\[ \text{Factors of 5} = 2 \times (19 + 3) = 44 \]
- For \(n = 98\), \(k = 3\):
\[ \text{Factors of 5} = 3 \times (19 + 3) = 66 \]
- For \(n = 97\), \(k = 4\):
\[ \text{Factors of 5} = 4 \times (19 + 3) = 88 \]
- Continuing this for all \(n\) down to \(1\), we will sum the contributions.

3. **Count the number of factors of 2:**
- Similarly, we count the factors of 2 for each \(n\) from 1 to 100.
- The contribution of \(n\) to the count of 2s is given by:
\[ \text{Contribution of } n = k \times \left\lfloor \frac{n}{2} \right\rfloor + k \times \left\lfloor \frac{n}{4} \right\rfloor + k \times \left\lfloor \frac{n}{8} \right\rfloor + \ldots \]

4. **Calculate contributions:**
- For \(n = 100\), \(k = 1\):
\[ \text{Factors of 2} = 1 \times (50 + 25 + 12 + 6 + 3 + 1) = 97 \]
- For \(n = 99\), \(k = 2\):
\[ \text{Factors of 2} = 2 \times (49 + 24 + 12 + 6 + 3 + 1) = 95 \]
- Continuing this for all \(n\) down to \(1\), we will sum the contributions.

5. **Determine the minimum of the two counts:**
- After calculating the total counts of factors of 5 and 2, we take the minimum of these two totals to find the number of trailing zeros.

6. **Final Calculation:**
- Let’s say we find that the total count of factors of 5 is \(X\) and the total count of factors of 2 is \(Y\).
- The number of trailing zeros in the product is given by:
\[ \text{Trailing Zeros} = \min(X, Y) \]