The number of zeros at the end of the following expression:
`P = {(2 xx 4 xx 6 xx 8 xx 10 xx .. 50) xx (55 xx 60 xx 65 xx 70 xx 75 xx …..100)}`

Options

A. less than 20
B. 57
C. 20
D. 36

Answer

To find the number of zeros at the end of the expression \( P = (2 \times 4 \times 6 \times 8 \times \ldots \times 50) \times (55 \times 60 \times 65 \times 70 \times 75 \times \ldots \times 100) \), we need to determine how many times the factors of 10 can be formed from the product. Each factor of 10 is made up of one factor of 2 and one factor of 5. Therefore, the number of zeros at the end of the product will be determined by the limiting factor, which is the number of times 5 appears in the factorization of the product, since there will be more factors of 2 than factors of 5.

### Step 1: Count the factors of 5 in the first part of the product
The first part of the product is \( 2 \times 4 \times 6 \times \ldots \times 50 \). This is the product of all even numbers from 2 to 50.

To find the number of factors of 5:
- The multiples of 5 in this range are: 10, 20, 30, 40, 50.
- Each contributes at least one factor of 5.

Counting these:
- 10 contributes 1 factor of 5.
- 20 contributes 1 factor of 5.
- 30 contributes 1 factor of 5.
- 40 contributes 1 factor of 5.
- 50 contributes 1 factor of 5.

Total from this part = 5 factors of 5.

### Step 2: Count the factors of 5 in the second part of the product
The second part of the product is \( 55 \times 60 \times 65 \times 70 \times 75 \times \ldots \times 100 \).

To find the number of factors of 5:
- The multiples of 5 in this range are: 55, 60, 65, 70, 75, 80, 85, 90, 95, 100.

Counting these:
- 55 contributes 1 factor of 5.
- 60 contributes 1 factor of 5.
- 65 contributes 1 factor of 5.
- 70 contributes 1 factor of 5.
- 75 contributes 2 factors of 5 (since \( 75 = 5^2 \times 3 \)).
- 80 contributes 1 factor of 5.
- 85 contributes 1 factor of 5.
- 90 contributes 1 factor of 5.
- 95 contributes 1 factor of 5.
- 100 contributes 2 factors of 5 (since \( 100 = 5^2 \times 4 \)).

Total from this part = 1 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 1 + 2 = 12 factors of 5.

### Step 3: Calculate the total number of factors of 5
Now, we combine the counts from both parts:
- From the first part: 5 factors of 5.
- From the second part: 12 factors of 5.

Total factors of 5 = 5 + 12 = 17.

### Step 4: Determine the number of trailing zeros
Since the number of trailing zeros is determined by the minimum of the number of factors of 2 and 5, and we have more factors of 2 than 5, the total number of trailing zeros in \( P \) is equal to the number of factors of 5.

Thus, the number of trailing zeros in \( P \) is **17**.

### Final Answer
The number of zeros at the end of the expression \( P \) is **17**.

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