Factorial Factoids

Copyright ©2003 by Paul Niquette. All rights reserved.
ne astonishing factoid about numbers is that the decimal ciphers are not equally represented among the most significant digits (see Benford's Law).  The bar-chart below summarizes the percentages of readily computed integers in which each decimal cipher appears as the most significant digit: Sophisticated solvers will notice something missing there.
 

The bars represent the percentages enjoyed by each decimal cipher as the most significant digit in populations of 90 computed integers (as indexed from n = 0, 1, 2, ... 89).  What the heck is going on with those factorials? 

n going all the way from n = 1 to n = 89, wherein...

89! = 16,507,955,160,908,461,081,216,919,262,453,619,
309,839,666,236,496,541,854,913,520,707,833,171,034,
378,509,739,399,912,570,787,600,662,729,080,382,999,
756,800,000,000,000,000,000,000
...the cipher 9 does not appear even once as the most significant digit.

Factorials are simply the products of consecutive integers, 1 times 2 times 3 times... up to some number n symbolized as n!  That's not an exclamatory statement.  The "!" was introduced in the early 19th century by the mathematician Christian Kamp and is pronounced "factorial."  By the way, expansion of n! is most often put the other way around: n! = n(n - 1)(n - 2)(n - 3)...3, 2, 1.  A "factoid" is something else.

Factorials do become humongous fast.  Consider the number of possible sequences you can deal after shuffling a deck of cards...

52! =  80,658,175,170,943,878,571,660,636,856,403,
766,975,289,505,440,883,277,824,000,000,000,000
...that's 8 bidecabillion permutations.  Moreover, to reach a googol, all you need is a mere 70!
 
In the late 18th century, a Scottish mathematician, James Stirling, unable to find batteries for his calculator, developed an approximation for factorials...
n! ~ (2 π n)1/2 (n / e)n
Thus, the product of the first n integers gets a formula a whole lot jazzier than that for the sum of the first n integers...
S(n) = n(n + 1)/2
Hey, the sum of the first n integers does not even have a name. 

Until now...Totorial.

As for a symbol, "&" might work ...
n& = n(n + 1)/2
...so that 36& = 666,  pronounced "36 totorial."
Christian Kamp would be pleased, don't you think?

Meanwhile, however huge n! gets, it does not seem to get a most significant digit of 9, as shown above.
 

Will that hold true for all factorials?

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