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Combinatorics Practice Problems and Solutions
The Number of Zeros at the End of 50! (50 Factorial)
This section is part of Mr. Stolyarov's Combinatorics Problems and Solutions.
The problem to be solved is as follows: How many zeros are at the end of 50! (50 factorial)? Note: 50! = 50*49*48*47*....*4*3*2*1Here, I offer two solutions.
First Solution: The multiples of ten which are components of 50! (without reusing any of the multiples) are (5*2), 10, 20, 30, 40, 50, (15*4), (25*6), (35*8), (45*12),
Multiplying these together, we have 1.63296*1017 = 163,296,000,000,000,000
There are 12 zeros in this number - which is the same number of zeros as in 50!, since no other factors of 50! can be multiplied together to produce multiples of ten. Thus, 50! has 12 zeros.
Second Solution: Each multiple of 5 - taken together with some even number - contributes a multiple of 10. Each multiple of 25, however - taken together with some even number - contributes two multiples of 10. So we can count the number of multiples of 5 in 50!. Then we can add another count of the number of multiples of 25 in 50!. There are 2 multiples of 25 in 50! (25 and 50) and 10 multiples of 5 (5, 10, 15, 20, 25, 30, 35, 40, 45, 50). 2+10 = 12. Thus, there are a total of 12 zeros in 50!
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 Right, Rebirth 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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