How many ways can 6 people be divided into 2 groups of 3?

How many ways can 6 people be divided into 2 groups of 3?

Correct Answer: 20

Step 1: Understand that we have 6 people and we want to divide them into 2 groups of 3.
Step 2: Calculate the total number of ways to arrange 6 people, which is 6! (6 factorial). This means 6 x 5 x 4 x 3 x 2 x 1 = 720.
Step 3: Since we are forming groups of 3, we need to account for the arrangements within each group. The number of ways to arrange 3 people in a group is 3! (3 factorial), which is 3 x 2 x 1 = 6.
Step 4: Since we have 2 groups of 3, we need to calculate the arrangements for both groups. This is (3! for the first group) x (3! for the second group) = 6 x 6 = 36.
Step 5: Now, we divide the total arrangements (720) by the arrangements within the groups (36) to avoid counting the same groups multiple times. So, we do 720 / 36 = 20.
Step 6: Finally, since the two groups are indistinguishable (Group A and Group B are the same as Group B and Group A), we divide by 2 to avoid double counting. So, we do 20 / 2 = 10.

Combinatorics – The problem involves calculating combinations to determine how many ways a set of people can be divided into groups.
Factorials – Understanding how to use factorials to calculate permutations and combinations is essential for solving the problem.
Overcounting – Recognizing that the order of groups does not matter, hence the need to divide by 2 to avoid overcounting.