How many ways are there to place 24 people into two treatment groups? How many ways are there to place 48 people into two treatment groups?
I thought the answer was (48 choose 24) because two treatment groups means that there will be 24 people in each group. However, the answer is supposed to be $1.6\times10^{13}$. What am I misunderstanding

You need to divide by 2 to get the desired answer.
E.g. Consider the set {A,B,C,D}. If I wanted to split them into two groups, I could do it like so:
Group 1: {A,B}
Group 2: {C,D}
OR
Group 1: {C,D}
Group 2: {A,B}
Despite being in different groups, you get the same outcome, and hence you divide by 2!, to avoid overcounting.
