Puzzle #44: Cookies!

Mom has been not been getting many treats lately. Having had enough of this, one fine day, she showed her hubby and three sons the door. “And take that fleabag cat with you!” She quickly locked and latched all doors and windows.

It’s cookie time!

Mom baked some cookies for herself. Oatmeal, yum!

1) There was at least one cookie in each of the seven combinations.
2) Sixteen cookies contained chocolate, fourteen contained raisins, and ten contained cinnamon.
3) Each of the seven combinations had a different number of cookies.
4) Seven cookies had chocolate but no raisins or cinnamon.
5) Each cookie had at least one of chocolate, raisins, and cinnamon.

How many did cookies did Mom enjoy, and how many were there of each combination?

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, April 9, 2014 at noon Pacific Time. I will post the answer shortly after.