The question goes:
A group of 60 second graders is to be randomly assigned to two classes of 30 each.
(The random assignment is ordered by the school district to ensure against any
bias.) Five of the second graders, Marcelle, Sarah, Michelle, Katy, and Camerin,
are close friends. What is the probability that they will all be in the same class?
Note that this question has been answered before in the two following links:
A group of 60 second graders is to be randomly assigned to two classes of 30 each…….
Probability that 4 friends will be in the same class
But I have a different issue to clarify for this question.
My idea is this:
Since 5 friends have to be together, then I can take it such that there are $60-5+1 = 56$ independent entities (other students and this group). From these 56, I will pick ${56 \choose 26}$ entities to be in one class, and ${56 \choose 30}$ in the other. Since ${56 \choose 26} = {56 \choose 30}$, I effectively should get a probability of $$2*\frac{{56\choose 26}}{{60\choose 30}} = \frac{126}{1121}$$
Yet I know this to be wrong. Can someone kindly point out to me where my flaw in logic is? Why does this train of thought not work?
Thank you!
Best Answer
If one of the "entities" is a group of $5$ students, and the other $55$ "entities" are individual students, then dividing them into a class of $26$ entities and a class of $30$ entities does not guarantee that there will be $30$ students in each class.
You might end up with $34$ students in the $30$-entity class: $29$ ordinary students and a $5$-student entity.