The answer is not 2

Computer Science Level 3

How many zeroes are found in $100!$ ?

Challenge: Do it without computer programming?

The answer is 30.

2 solutions

Arulx Z
Nov 14, 2015

Python one-liner

1	`str(reduce(lambda x, y : x * y, [1] + range(1, 101))).count('0')`

Moderator note:

Simple standard approach.

@Calvin Lin how to do this without programming (as asked by the OP).

Arulx Z - 5 years, 6 months ago

There isn't an easy way to count the number of digits in a "random" large number. It has to be done essentially by hand.

Calvin Lin Staff - 5 years, 6 months ago

Check this out. As Calvin said, there is no closed form solution for this like the trailing zeros case. There are a few approximations at best but they all show weak convergence.

Prasun Biswas - 5 years, 6 months ago

Thanks for the link. It was really helpful.

Arulx Z - 5 years, 6 months ago

Navneel Mandal
Nov 8, 2015

I have done it with computer programming.(Sorry!) This is my code: #quite long, but works!

Count=0

fact=1

for ctr in range(1,101):

fact*=ctr

m=str(fact)

for x in m:

    if x=="0":

        Count+=1

    else:

        continue

print "The total zeroes are",Count

@Hobart Pao , in response to the challenge, I don't think there's a definite closed form to give the number of zeros in $n!$ for non-negative integer $n$ .

Source .

There are a few approximations for this given in the answers section of the above-mentioned source post but they all show weak convergence.

Prasun Biswas - 5 years, 7 months ago

The answer is not 2

The answer is 30.

2 solutions

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