# Combinatorics Problems and Solutions

G. Stolyarov II
Spring 2008 - Republished July 6, 2014

1. The Number of Zeros at the End of 50! (50 Factorial)

2. The Number of Zeros at the End of 1000! (1000 Factorial)

We wish to find the number of zeros at the end of 1000! = 1000*999*998*997*....*4*3*2*1

We examine the factors of 1000! to see which can be used to produce multiples of 10.

For 1000!, each multiple of 5 - taken together with some even number - contributes a multiple of 10. Each multiple of 25 - taken together with some multiple of 4 - contributes two multiples of 10. Each multiple of 125 - taken together with some multiple of 8 - contributes three multiples of 10.  Each multiple of 625  - taken together with some multiple of 16 - contributes four multiples of 10.

We can count each multiple of 5 in 1000! once. Then we can count each multiple of 25 in 1000! once in order to altogether count each multiple of 25 twice. Then we can count each multiple of 125 in 1000! once in order to altogether count each multiple of 125 thrice. There is also one multiple of 625 - i.e., 625 itself, in 1000!, which we count once in order to count the multiples of 625 altogether four times.

There are 1000/5 = 200 multiples of 5 in 1000!

There are 1000/25 = 40 multiples of 25 in 1000!

There are 1000/125 = 8 multiples of 125 in 1000!

There is also one multiple of 625 - i.e., 625 itself, in 1000!

Adding them together, we have 200+40+8+1 = 249 zeros at the end of 1000!.

4. Recurrence Relations and Generating Functions: Problem and Solution I

5. Partitions of Integers: Practice Problems and Solutions - Part I

Gennady Stolyarov II (G. Stolyarov II) is an actuary, science-fiction novelist, independent philosophical essayist, poet, amateur mathematician, composer, and Editor-in-Chief of The Rational Argumentator, a magazine championing the principles of reason, rights, and progress

In December 2013, Mr. Stolyarov published Death is Wrong, an ambitious children’s book on life extension illustrated by his wife Wendy. Death is Wrong can be found on Amazon in paperback and Kindle formats.

Mr. Stolyarov has contributed articles to the Institute for Ethics and Emerging Technologies (IEET), The Wave Chronicle, Le Quebecois Libre, Brighter Brains Institute, Immortal Life, Enter Stage RightRebirth of Reason, The Liberal Institute, and the Ludwig von Mises Institute. Mr. Stolyarov also published his articles on Associated Content (subsequently the Yahoo! Contributor Network) from 2007 until its closure in 2014, in an effort to assist the spread of rational ideas. He held the highest Clout Level (10) possible on the Yahoo! Contributor Network and was one of its Page View Millionaires, with over 3.1 million views.

Mr. Stolyarov holds the professional insurance designations of Associate of the Society of Actuaries (ASA), Associate of the Casualty Actuarial Society (ACAS), Member of the American Academy of Actuaries (MAAA), Chartered Property Casualty Underwriter (CPCU), Associate in Reinsurance (ARe), Associate in Regulation and Compliance (ARC), Associate in Personal Insurance (API), Associate in Insurance Services (AIS), Accredited Insurance Examiner (AIE), and Associate in Insurance Accounting and Finance (AIAF).

Mr. Stolyarov has written a science fiction novel, Eden against the Colossus, a philosophical treatise, A Rational Cosmology,  a play, Implied Consent, and a free self-help treatise, The Best Self-Help is Free. You can watch his YouTube Videos. Mr. Stolyarov can be contacted at gennadystolyarovii@yahoo.com.

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