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Competition

Why should we work together considering that we are going to be competing for admission in the same colleges?

Competition is the correct approach in a zero sum game. If your win is same as my loss, I should make every effort so you lose and as a consequence I win. Cooperation is a correct approach if there is a possibility that we both can win or we both can lose. There is no reason to believe that a college counsellor has a quota for the participants of this group and they will pick a certain number of kids from this group. Chances are that if they like all of us, they will offer it to all of us. If we all made avoidable mistakes, chances are we will receive "thank you" letters.