Thursday, November 02, 2006

 

stirling's formula for factorials

Unbelievably large numbers are sometimes the answers to innocent looking questions.For instance,imagine that you are playing with an ordinary deck of 52 cards.As you shuffle and re-shuffle the deck you wonder:How many ways could the deck be sheffled?That is how many different ways can the deck be put in order?You reason that there are 52 choices for the first card,then 51 choices for the second card,then 50 for the third,etc.This gives a total of
52x51x50x......2x1.
ways to order a deck of cards.We call this number "52 factorial"and write it as the numeral 52 with an exclamation point:52! this number turns out to be the 68 digit monster

which means that if every one on earth shuffled cards from now until the end of the universe,at a rate of 1000 shuffles per second,we wouldn't even scratch the surface in getting all possible orders.Whew!No wonder we use exclamation marks!

For any positive integer n we calculate "n factorial" by multiplying together all integers up to and including n, that is,

n!=1x2x3x....xn

1!=1 ; 2!=2 ; 3!=6 ; 4!=24 ; 5!=120

6!=720 ; 7!=5040 ; 8!=40320 ; 9!=362880 ; 10!=3628800

STIRLING'S FORMULA

Factorials start off reasonably small, but by 10! we are already in millions, and it doesn't take along until factorials are unwieldly behemoths like 52! above.Unfortunately there is no shortcut formula for n! ,you have to do all of the multiplication.On the other hand,there is a famous approximate formula,named after the Scottish mathematician James Stirling(1692-1770),that gives a pretty accurate idea about the size n!

n!=√2∏n*(n/e)ˆn


# posted by Unknown @ 5:04 AM
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