Math Is Fun Forum

John E. Franklin

Member

Registered: 2005-08-29

Posts: 3,588

3 people sharing discrete identical objects

Given m identical objects such as evenly
cut slices of an apple pie, how can you
distribute them to Pat, Mike, and John,
providing everyone gets one or more
pieces?

Possible answer based on doing examples
on paper:
   (m-1)(m-2) / 2 ways     [this formula only works for 3 unique people]
                                    [and the formula is not proven at all]
                                [and the formula is a triangular number too]
If m is 3, then ways = 1, and
each person gets 1 pieces.

There is a variation; think  600 instead of  711, because 600 + 111 = 711
This variation says that for 711 or 7+1+1 or 9 pieces of
pie and 3 people where 1 or more is given to each, then
m = 9, so  ways is (8)(7)/2 = 28 ways, but the
variation is that the answer and formula are the same
for 600, or 6 + 0 + 0, or 6 pieces of pie shared by
3 people where zero pieces is allowed for up to 2 of the
people, but they must eat the whole pie.
So to compute, the 600 goes to 711, and the m=9 is
still used.  Still 28 ways.
And also if the range of pieces of pie was 2 to 8 instead of
1 to 7 or 0 to 6, I think the same formula can be used, because
when you write out the combinations where people are identical,
then the numbers are all the same between these scenarios
except you just add 111 to get to the other scenarios:
600  711  822...

Next I'm gonna see what 4 people sharing pie comes up with
for a formula, but I can't prove it because I'm dumb.

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igloo myrtilles fourmis