Hi,

Oops, For files of K bits, those files that can be mapped to a smaller files
should be (1-2^(1-K))*2^K and the remaining files are 2^(1-K)*2^K.  Both
these number should sum to 2^K.

Change 2^K to K in the earlier posting.

This expression are evaluated from the series repeated here again

Number of files of size one bit less that K  =2^(K-1)=2^(-1)(2^K)
Number of files of size two bit less that K  =2^(K-2)=2^(-2)(2^K)

... and so on ....

Number of files of size 1 bit
=2^(K-(K-1))=2^(1-K)(2^K)

The total number of files of size smaller than K bits is the sum of the
series times 2^K (ie. sum of a series times the sum of all K bits files)

 (  2^(-1)+ 2^(-2)+............+2^(1-K) )  *  (2^K)

Sum of the series is 2^(-1)( 1 - 2^((-1)(K-1))   sum of a geometric series
                                   --------------------------------   of
common ratio 2^(-1)
                                      1 - 2^(-1)
which evalutes to 1-2^(1-K).  Which gives us the number of all files of size
smaller than K bits as (1-2^(1-K))*2^K.

As for mapping one K bits file to file of size ZERO bit; well, where are you
going to store this file?  And how are going to share with someone?

Dale King <KingD@tce.com> wrote in message
news:<38B5A4E9.7773AACF@tce.com>...
> Care to explain how you arrived at those formulas? They are obviously
> wrong because for instance as k gets larger you can compress all the
> files of k bits except for a miniscule fractional file that goes rapidly
> to zero as k gets larger (at k=5, Excel gives me 2^k).
..............