When dividing 6 students into 3 groups, how many ways can the groups be arranged if each group is assigned a unique topic?

To find the number of ways to divide 6 students into 3 groups, where each group is assigned a unique topic, we approach the problem in a few steps.

First, we will divide the 6 students into 3 distinct groups. The groups can have different sizes, and since the assignment of unique topics indicates that the groups are distinguishable, we will treat them as such.

1. Choosing Students for Each Group:

The arrangement starts with deciding how many students go into each group, ensuring that all 6 students are part of the distribution. For example, if we assign 2 students to the first group, 2 students to the second group, and 2 students to the third group, we use the multinomial coefficient for this distribution.

2. Calculating the Group Combinations:

The number of ways to choose the first group of 2 students from 6 can be defined as "6 choose 2". After choosing the first group, 4 students remain. Next, we choose 2 students from these 4 for the second group, which is "4 choose 2". The final 2 students automatically form the third group. The binomial coefficients for these choices are