Mary and I or Mary and me?

A problem of case

Mary and I are delighted to be here today.(NOT Mary and me)
The letter was addressed to Mary and me.(NOT Mary and I)
In 1, "Mary and I" are subjects, which is why the pronoun takes the subjective case ("I").
In 2, "Mary and me" are objects, which is why the pronoun takes the objective case ("me").
An easy way to check the correct case is to try the sentence without Mary.
Would you say "I am delighted to be here" or "Me am delighted to be here"?
Would you say "The letter was addressed to me" or "The letter was addressed to I"?

http://www.englishclub.com/grammar/pronouns-case.htm