Source: DiscreteMath Question posed on mathforum.org (9/22/2009) by "naslund19"

**Hint:** According to the poser of this problem, the key is to use the Principle of Inclusion-Exclusion.

If that does not help, you might try it with smaller cases, e.g. three groups of three students, etc....adjusting the number of rows and columns as needed.

**Solution Commentary:** When "naslund19" posed this question, only "Ben" responded (9/22/2009):

"Not sure what "inclusion-exclusion" has to do with this."

"With 6 rows, what is the probability that any one group will be in its original row? So what's the probability a group is __not__ in its original row? So (1)-(3) and first part of (4) look like straight 'binomial' problems.

For the second part of (4), what is the probability that any one student is in his/her original column? So what's the probability (s)he is __not__ in that column? How do we find the probability that EACH student is not in his/her original column?"

"Finally, how do we combine the two probabilities in (4) to get the probability no group is in its original row **and** no student is in her/his original column?"

"Thanks for a great problem!"

Now, that was Ben's response....do you agree?