How Many Trailing Zeros Does 100! Have?

Step-by-Step Explanation
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Determining the Number of Trailing Zeros in 100!

a) 10

b) 20

c) 24

d) 100

e) 8

To find out how many trailing zeros are in the number 100!, we need to understand how these zeros are formed. In the multiplication of integers, trailing zeros are created by factors of 10. Since 10 can be decomposed into 5 and 2, we need to find how many pairs of these factors are present in 100!.

Breaking Down the Problem
In the factorial 100!, we need to count the number of 2s and 5s that appear. However, since there are usually more factors of 2 than 5 in any sequence of consecutive numbers, the number of trailing zeros will be determined by the number of 5s.

Counting the Factors of 5
To find out how many times 5 is a factor in 100!, we need to count every multiple of 5 within this range:

Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100. This gives us 20 multiples of 5.

Multiples of 5^2 (25): 25, 50, 75, 100. Each of these multiples contributes an additional factor of 5. Thus, we get 4 extra factors of 5 from these numbers.

Summing Up the Factors
From the multiples of 5, we have 20 factors. From the multiples of 525^252, we get an additional 4 factors, making a total of 24 factors of 5 in 100!.

Conclusion
Since each pair of 2 and 5 forms a trailing zero, and there are plenty of 2s to pair with each 5, the number of trailing zeros in 100! is equal to the number of 5s. Therefore, 100! has 24 trailing zeros.

ANSWER C.

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