I've seen a bunch of questions about dividing a group of $N$ into groups of a specified size, but I am unsure about how to calculate the total number of ways to divide a group of $N$ people into $2$ distinct groups..
The questions states that one group could be empty, and that a group could have sizes from $0, 1, 2, …, N$.
The question then goes on to ask what is the probability that one of the groups has exactly $3$ people in it. I presume this would be calculated by dividing $N\choose 3$ by the total number of ways calculated above, but any other comments would be greatly appreciated!
Best Answer
Suppose you lined every one of them up, and you could assign everyone a $0$ or $1$, for either group. Then each person could either have a $0$ or $1$. So for $N$ people, there are $2^N$ ways of doing this so $2^N$ different groups could be formed. Now to find the probability that one of these groups is of size $3$, how many ways can you pick $3$ people from $N$? Knowing this, you can calculate the probability.