Six Billion Factorial

Date: 05/15/2003 at 23:45:14
From: DJ
Subject: Permutations

6 billion p 100

I know how to do permutations but there must be a quick way to 
do a problem like this. Even my calculator lists error when I try to 
do it. Is there a formula that I don't know about?

6000000000!/6000000000-100!

Date: 05/16/2003 at 10:53:46
From: Doctor Tom
Subject: Re: Permutations

Hi DJ,

The problem is that your calculator can only work with numbers up to 
some given size, and 6B p 100 is too big for it.

It is approximately (6,000,000,000)^100, since the other numbers in 
the product are pretty close to 6 billion. This would be a number
with about 977 digits - in fact, it's approximately 6.53x10^977.

- Doctor Tom, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/16/2003 at 12:04:47
From: Doctor Stephen
Subject: Re: Permutations

Hi DJ,

People will tell you that there is no number that describes infinity 
but I believe 6000000000! is getting somewhere close. If you try to 
calculate the answer to the above question by finding the answer to 
the top and bottom of the fraction separately, you will have problems.

The actual solution arises from manipulating the entire fraction to 
make it more calculator-friendly. We know that six billion factorial 
is the same as multiplying all the numbers up to six billion together. 
The bottom line of the fraction is then the same as multiplying all 
the numbers up to six billion minus one hundred together.

  6 billion p 100 then becomes 6000000000!/5999999900!

Lets imagine we have a simpler case for a moment, say 6 p 3. The 
number of permutations in this case is:

          (6 * 5 * 4 * 3 * 2 * 1) / (3 * 2 * 1)

From this we can cancel numbers that appear on both the top and bottom 
lines of the fraction, leaving the answer to be 6 * 5 * 4. The same 
method can be applied to the more difficult case involving large 
numbers. The first 5999999900 terms can be removed from the top and 
the bottom of the fraction leaving just 100 terms, that is:

  6000000000 * 5999999999 * 5999999998 * ... * 5999999901

I tried this on my small Casio calculator and got an error because the 
answer is still very large. The difference in doing the calculation 
this way is that computer software such as Microsoft Excel and most 
graphical calculators can cope with 100 terms of this magnitude but 
cannot calculate 6000000000! on its own.

Giving the calculation a try in a computer mathematical program, I got 
an answer around 10 to the power of 1000.

- Doctor Stephen, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/16/2003 at 12:25:46
From: Doctor Ian
Subject: Re: Permutations

Hi DJ,

To add to what Drs. Tom and Stephen have already said, this is one of
those situations where scientific notation and the properties of
exponents can be useful. If, as Dr. Stephen suggests, we've reduced
the problem to computing 

  6,000,000,000 * 5,999,999,999 * ... * 5,999,999,901

we can note that this is about equal to 

  (6 x 10^9) * (5.999999999 x 10^9) * ... * (5.999999901 x 10^9)

which can be rewritten

  (6 * 5.999999999 * ... * 5.999999901) * (10^9)^100

which, as Dr. Tom points out, is about equal to 

    6^100 * (10^9)^100

So, how do you find 6^100?  If you have a calculator, you just use the
y^x button to get

  6^100 ~ 6.5 x 10^77      [where '~' means 'approximately equals']

and then you can combine exponents:

  (6.5 x 10^77) * (10^900) = 6.5 x 10^977

But what if you don't have a calculator handy (or the batteries have
died)?  One way is to start computing powers of 6 until you get close
to a power of 10.  It turns out that 

  6^9 = 10,077,696

So we can approximate 6^9 as 10^7.  Then

  6^100 = 6^(99+1)

        = 6^99 * 6^1

        = (6^9)^11 * 6

        ~ (10^7)^11 * 6

        = 10^77 * 6

which might be good enough for government work, as they say.  

By the way, this kind of problem is a classic example of why it's
important to learn to use calculators as an amplifier, but not as a
crutch:

   Are Calculators Smart?
   http://mathforum.org/library/drmath/view/57030.html 

I hope this helps.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/