Introduction:

Calculating the number of zeros in a factorial can be a challenging task, but there is a formula that can simplify the process. In order to understand this formula, let's first discuss the concept of zeros in a factorial.

Understanding Zeros in Factorials:

When we calculate the factorial of a number, we multiply all the positive integers up to that number. For example, 5! (read as "5 factorial") equals 5 x 4 x 3 x 2 x 1, which is equal to 120.

The number of zeros at the end of a factorial depends on the number of times the factorial is divisible by 10. Since 10 is the product of 2 and 5, the number of zeros in a factorial is determined by the number of factors of 2 and 5 in the factorial.

Factors of 2 and 5:

In any factorial, the number of factors of 2 is always greater than or equal to the number of factors of 5. Therefore, we only need to count the number of factors of 5 to determine the number of zeros.

Formula:

The formula to calculate the number of zeros in a factorial is:

Number of Zeros = (n/5) + (n/25) + (n/125) + ...

Where 'n' is the number whose factorial is being calculated.

Explanation:

Let's take an example to understand the formula better. We'll calculate the number of zeros in 100!.

1. n = 100
2. Number of Zeros = (100/5) + (100/25) + (100/125) + ...
3. Number of Zeros = 20 + 4 + 0.8 + ...
4. Number of Zeros = 24.8 (rounded down to 24)

Therefore, there are 24 zeros at the end of 100!.

Conclusion:

The formula (n/5) + (n/25) + (n/125) + ... can be used to calculate the number of zeros in a factorial. By counting the factors of 5 in the factorial, we can determine the number of zeros at the end. Remember to round down the final result to obtain the exact number of zeros.