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Let's analyze the problem step-by-step to answer the question.

### Summary of Information Given:

1. **Horses Available:** There are 10 horses: Horse-1, Horse-2, Horse-3, Horse-4, Horse-5, Horse-6, Horse-7, Horse-8, Horse-9, and Horse-10.
   
2. **Chestnut-Coloured Horses:** Horse-1, Horse-3, Horse-7, Horse-8, and Horse-9.
   
3. **Black-Coloured Horses:** Horse-2, Horse-4, Horse-5, Horse-6, and Horse-10.

4. **Condition on Selection:** Mr. Davidson needs to select **6 horses** such that there are an **equal number of chestnut and black-coloured horses**.

### Constraints:
The following conditions must be respected when selecting the horses:

1. **Condition 1:** If **Horse-2** is in the group, then at most **one among Horse-3 and Horse-6** can be in the group.
   
2. **Condition 2:** If **Horse-1** is in the group, then **neither Horse-8 nor Horse-9** can be in the group.
   
3. **Condition 3:** Among **Horse-4, Horse-5, and Horse-6**, **at least one must be in the group**, and **at most two** of them can be in the group.
   
4. **Condition 4:** If **Horse-5** is in the group, then **neither Horse-2 nor Horse-10** can be in the group.
   
5. **Condition 5:** If **Horse-9** is in the group, then **Horse-4 must be in the group**.

### Goal:
Determine the number of ways Mr. Davidson can select a group of 6 horses that meet all these conditions, with an equal number of chestnut and black-coloured horses (i.e., 3 chestnut and 3 black).

### Approach:

1. **Select Combinations of Chestnut Horses:** We need to pick exactly **3 chestnut horses** from the available five (Horse-1, Horse-3, Horse-7, Horse-8, Horse-9).
   
2. **Select Combinations of Black Horses:** Simultaneously, we must pick **3 black horses** from the available five (Horse-2, Horse-4, Horse-5, Horse-6, Horse-10).

3. **Apply Constraints:** We need to apply each constraint to filter out any invalid combinations.

Given the complexity of this problem, enumerating all combinations while applying each constraint step-by-step is necessary to find the valid selections.

I will proceed with calculating possible combinations based on these rules and will let you know the solution shortly.The number of valid ways Mr. Davidson can select a group of 6 horses, ensuring an equal number of chestnut and black-coloured horses while satisfying all given constraints, is **11**.

Would you like further details on the specific valid combinations or any explanation on the approach?

Here are some related questions that may expand your understanding:

1. How would the result change if Mr. Davidson needed an unequal number of chestnut and black-coloured horses?
2. What alternative approach could be used to solve this type of constraint-based selection problem?
3. How would adding or removing horses from the initial selection impact the solution?
4. What if one of the constraints, such as Condition 3, was modified or removed? How would that affect the result?
5. Can this approach be applied to non-race-related selection problems with similar constraints?

**Tip:** When dealing with multiple constraints, breaking down each condition and using systematic checks can simplify the solution process.

Mr. Davidson, a horse race organiser, wants to select a group of six horses from ten horses in a stable with constraints on the selection based on color and additional conditions on group composition. He needs to ensure there are equal numbers of chestnut and black-coloured horses among the selected group. How many ways can Mr. Davidson select the group of six horses that meet these conditions?

Explore how to solve a combinatorial selection problem where a group of six horses must be chosen from a stable of ten under specific color and constraint-based conditions. Learn about the application of combinations, logical constraints, and set theory. Suitable for students in Grades 10-12.

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