Find the number of zeroes in the product
`1^(1)xx2^(2)xx3^(3)xx4^(4)xx5^(5)xx6^(6)xx…xx49^(49)`

Answer

To find the number of zeros in the product \(1^{1} \times 2^{2} \times 3^{3} \times \ldots \times 49^{49}\), we need to determine how many times 10 is a factor in this product. Since \(10 = 2 \times 5\), we need to find the minimum of the number of factors of 2 and the number of factors of 5 in the product.

### Step 1: Count the factors of 5

The first step is to count how many times 5 appears as a factor in the product. We can do this by considering the contributions from each term \(n^n\) where \(n\) is a multiple of 5.

1. **Identify multiples of 5**: The multiples of 5 from 1 to 49 are \(5, 10, 15, 20, 25, 30, 35, 40, 45\).
2. **Count contributions**:
- For \(5^5\), it contributes 5 factors of 5.
- For \(10^{10}\), it contributes 10 factors of 5.
- For \(15^{15}\), it contributes 15 factors of 5.
- For \(20^{20}\), it contributes 20 factors of 5.
- For \(25^{25}\), it contributes 25 factors of 5.
- For \(30^{30}\), it contributes 30 factors of 5.
- For \(35^{35}\), it contributes 35 factors of 5.
- For \(40^{40}\), it contributes 40 factors of 5.
- For \(45^{45}\), it contributes 45 factors of 5.

Now, we sum these contributions:
\[ 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 45 = 225 \]

### Step 2: Count the factors of 2

Next, we count how many times 2 appears as a factor in the product.

1. **Identify multiples of 2**: The multiples of 2 from 1 to 49 are \(2, 4, 6, \ldots, 48\).
2. **Count contributions**:
- For \(2^2\), it contributes 2 factors of 2.
- For \(4^4\), it contributes 8 factors of 2.
- For \(6^6\), it contributes 12 factors of 2.
- For \(8^8\), it contributes 24 factors of 2.
- For \(10^{10}\), it contributes 20 factors of 2.
- For \(12^{12}\), it contributes 24 factors of 2.
- For \(14^{14}\), it contributes 28 factors of 2.
- For \(16^{16}\), it contributes 64 factors of 2.
- For \(18^{18}\), it contributes 36 factors of 2.
- For \(20^{20}\), it contributes 40 factors of 2.
- For \(22^{22}\), it contributes 44 factors of 2.
- For \(24^{24}\), it contributes 48 factors of 2.
- For \(26^{26}\), it contributes 52 factors of 2.
- For \(28^{28}\), it contributes 56 factors of 2.
- For \(30^{30}\), it contributes 60 factors of 2.
- For \(32^{32}\), it contributes 128 factors of 2.
- For \(34^{34}\), it contributes 68 factors of 2.
- For \(36^{36}\), it contributes 72 factors of 2.
- For \(38^{38}\), it contributes 76 factors of 2.
- For \(40^{40}\), it contributes 80 factors of 2.
- For \(42^{42}\), it contributes 84 factors of 2.
- For \(44^{44}\), it contributes 88 factors of 2.
- For \(46^{46}\), it contributes 92 factors of 2.
- For \(48^{48}\), it contributes 96 factors of 2.

Now, we sum these contributions:
\[ 2 + 8 + 12 + 24 + 20 + 24 + 28 + 64 + 36 + 40 + 44 + 48 + 52 + 56 + 60 + 128 + 68 + 72 + 76 + 80 + 84 + 88 + 92 + 96 = 576 \]

### Step 3: Determine the number of zeros

The number of trailing zeros in the product is determined by the minimum of the number of factors of 2 and the number of factors of 5:
\[ \text{Number of zeros} = \min(225, 576) = 225 \]

Thus, the final answer is:
\[ \boxed{225} \]