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Let's solve the given problem step by step.

### Part (a): Two Equal Groups

You want to divide a group of six people into two equal groups of 3 people each.

1. **Step 1: Select 3 people out of 6.**  
   The number of ways to choose 3 people from 6 is given by the combination formula \( \binom{6}{3} \):
   \[
   \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
   \]

2. **Step 2: Adjust for indistinguishable groups.**  
   Since the two groups are indistinguishable, we have counted each division twice (once as Group A vs Group B, and once as Group B vs Group A). So, we divide by 2:
   \[
   \text{Total ways} = \frac{20}{2} = 10
   \]

Thus, there are **10 ways** to divide the six people into two equal groups.

### Part (b): Two Unequal Groups

Now, you want to divide the group of six people into two unequal groups such that each group has at least one person.

Possible group sizes:
- One group of 1 person and another group of 5 people.
- One group of 2 people and another group of 4 people.

Let's calculate these cases:

1. **Case 1: One person in the first group and five in the second.**
   \[
   \text{Number of ways} = \binom{6}{1} = 6
   \]

2. **Case 2: Two people in the first group and four in the second.**
   \[
   \text{Number of ways} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15
   \]

3. **Case 3: Three people in the first group and three in the second.**
   \[
   \text{This is already calculated in part (a)} \rightarrow 10 \text{ ways, but we do not consider this here because the groups are supposed to be unequal.}
   \]

Thus, the total number of ways is \( 6 + 15 = 21 \).

Therefore, there are **21 ways** to divide the group of six people into two unequal groups with at least one person in each group.

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Would you like more details or have any questions?

Here are five related questions you might find interesting:

1. How many ways can you divide 8 people into two equal groups?
2. If you had 7 people, how many ways could you divide them into two unequal groups?
3. How would the calculation change if the groups were distinguishable?
4. What if the problem asked for 3 unequal groups instead of 2?
5. How can these combinatorial problems be applied in real-world situations?

**Tip:** When dividing into groups, always consider whether the groups are distinguishable or not; this impacts how you count the possible divisions.

The query submitted by the user and the corresponding solution: How many ways can you divide 6 people into two equal and unequal groups?

Explore how to calculate the number of ways to divide a group of 6 people into two equal groups and two unequal groups. This problem involves combinatorial principles and the combination formula. Suitable for students in Grades 7-9.

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