Need help with an arrangement problem of a word

[Sunohara]

Consider the word AEEMRRY, how many arrangements are there without the words "eye" and "ram"?

The answer in my textbook is 7!/2!2! - 5!/2! - 5!/2! + 3!, but I don't understand where the 3! came from.. I understand that 7!/2!2! is the total number of arrangements and 5!/2! is basically the arrangements of "eye" and the next 5!/2! is the arrangements of "ram" in the word. However I don't understand why the book is adding 3!.

My original answer was 7!/2!2! - 5!/2! - 5!/2!, so it would be very great if someone can explain why there is a +3!. My teacher hinted towards the class that it was because there was an overlap or something, but I don't see where the overlap is occurring. Then my friend said that it was because we add 3! because we are over counting, but I don't understand that too...(Angry)

 

[Valkarie]

The 3! comes from the arrangements of the overlapping letters. In the word AEEMRRY there are three letters that are repeated - M, R, and Y. The 7!/2!2! is the total number of arrangements, the 5!/2! is the arrangements of "eye" and the next 5!/2! is the arrangements of "ram" in the word. However, when we subtract the two 5!/2!'s, we are removing all the arrangements of the overlapping letters (M, R and Y) twice. So, to make sure we only count each arrangement of the overlapping letters once, we need to add 3! (the number of arrangements of the overlapping letters).