The tuition exchange program, which @happymomof1 cites above, is definitely something I should have mentioned in my "Ask the Dean" response. BUT ... chances are good that it wouldn't be the right option for the boy who sent the initial question.
The Tuition Exchange Program offers free tuition to children of employees at participating colleges. However, the vast majority of member schools are not the "elite" colleges to which this guy aspires. And the competition to be named a TEP recipient at ANY school is fierce. Moreover, TEP doesn't typically include room and board, and many TEP members cap the tuition funds they provide at $35,000 per year, which doesn't cover full tuition at the handful of private colleges on the TEP list that this kid might consider.
Thus, if this boy views himself as Ivy-eligible and doesn't want to attend his father's college, he would probably more satisfied by snagging a merit scholarship at a strong non-Ivy (e.g., Emory, BC, Tulane, ND, Wash U, and many more ... ) than by aiming for a TEP award.
So, in this student's case, TEP would be a red herring, but it's certainly worth investigating for OTHER students in similar shoes ... those who have a parent working at a member school who won't spring for tuition elsewhere.
@LiLiBug - I feel your son's pain, but it is definitely not the end of the world. College admission officials understand that--with the exception of Early Decision candidates--students apply to more than one college. And they understand, too, that this process can be very stressful and confusing so mistakes like your son's are bound to happen sometimes.
But it still would be wise for your son to do a little damage control. He should send an email to his regional rep* at the three colleges that received the "wrong" statement. He can explain that he is very enthused about that college and had planned to write an "Additional Information" statement that was specific to each of the colleges on his very short list, not realizing that--when he sent the first one--it would go to all colleges. Thus, in every email, he should include a new statement that is customized for each college, asking the admission staff to overlook the one on the Common Application. He should also apologize for the mistake and for making them take extra time during a hectic period.
If your son is not admitted to these colleges, it won't be because of this snafu so do tell him to take a deep breath and stop worrying about it.
*if you don't know the name and email address, and if you can't find it online, call the admission office and ask.
@thinkingfastslow -This is a very minor error and, no matter how you handle it, it won't have any negative impact on your Princeton outcome. But it's fine to send an email to your Princeton regional rep to explain why they will see "Out of district" on your transcript. You can name the school and the specific class you took, if it's not clear from the transcript. Since you're writing anyway, it would be appropriate to explain very briefly WHY you took this summer class. For instance, did it help you to accelerate and jump to a higher math class in the fall?