Q:

P equals NP description is easy

In the official link of the P=NP problem at the below page:
P=NP Problem Official Description

, there is mentioned a problem of organizing housing accommodations for a group of four hundred university students and where the hostel capacity is 100 at max and also the dean has given a list of pairs of students who are 'incompatible', that is, they can't be accomodated together. The problem further states that building such a list from scratch would be difficult/hard. But, I did some thinking and came up with below solution:

We proceed by counting the exact number of pairs possible which can be done as follows: 
1st we select 100 students out of 400 available students.=(400C100) ways. Next, we select 2 students out of each of the 100 students' set in 100C2 ways. Then for each of the set it is "added" to the final result set if that pair does not exists in the dean's pair of incompatible set.

I know that this process of making a list will take a long time as the list itself is very long. But, if we consider the 1st combination set of 100 students out of 400 students for the sake of LUCK for the few lucky students( I mean if it is sort of a lucky draw and only the 1st set of 100 students is finally selected), then it will be an easy problem altogether!
Anyways, we can 'tell' in this problem that solution set exists or not as we can directly say if any pairs are possible by looking at the dean's list of incompatible students as all those and only all those should not be in the result set so if there are 50 such pairs that means there are 100 incompatible students and we can select the 1st 50 from this list and other 50 from remaining of the 350 students? This does not seem 'HARD' as the 3SAT problem where just telling that a solution exists is enough. (As in this case the total number of combinations are 400C100 * [  100C2  - the number of pairs that are in the selected list and in the dean's list also]
(This can also be written as 400C100 * [  100C2 MINUS pairs in the dean's list], where MINUS is the operation in the database terminology.
 Please shed some light on my research and if I am right or wrong?

Thanks

A:

I got the answer in one of the comments to below question:
Math Stack Exchange same topic question

The dorm is not a 50 bed dorm but a 100 bed dorm and it is not the case that 2 people can share a bed and incompatible students cannot be on the same bed but still stay in the dormitory in separate beds.

