Day 1: Partridges in a Pear Tree

The partridge in the pear tree was feeling lonely so it decided to invite 7 of its partridge friends to join it for a festive dinner party. Of course, the dessert is poached pears in syrup, freshly picked from the tree.

The partridge host has prepared 12 pears to serve. She does not eat any pears. However her 7 guests are quite particular:

  1. Every guest must have at least one pear.
  2. Two of the guests are particularly important, so they both must have at least two pears.
  3. No guest may have four or more pears (otherwise they will feel insulted for being greedy).

How many ways are there to distribute all the pears?

All the guests are distinguishable (so order matters) and yes, all the pears are served (whole).

This problem is part of the Advent Calendar 2015.

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