The integers from 1 through 10 (inclusive) are divided into three groups, each containing at least one number. These groups satisfy the additional property that if x is in a group and 2x≤10, then 2x is in the same group. How many different ways are there to create the groups
To solve this problem, we can use a recursive approach. We start by assigning the number 1 to any one of the three groups. Since the three groups must each contain at least one number, we have three options for the first group.
Now, for each subsequent number from 2 to 10, we have to decide which group to place it in. According to the given condition, if a number x is in a group and 2x ≤ 10, then 2x must also be in the same group.
Let's consider each number one by one and decide which group it can be placed in:
- For the number 2, since 2x ≤ 10, we can place it in the same group as 1.
- For the number 3, since 2x ≤ 10, it can't be in the same group as 1. So, there are two possible groups left where it can be placed.
- For the number 4, since 2x ≤ 10, we can place it in the same group as either 2 or 1.
- For the number 5, since 2x ≤ 10, it can't be in the same group as 1 or 2. So, there is only one possible group left where it can be placed.
- For the number 6, since 2x ≤ 10, we can place it in the same group as either 3 or 2.
- For the number 7, since 2x ≤ 10, it can't be in the same group as 2 or 3. So, there is only one possible group left.
- For the number 8, since 2x ≤ 10, it can't be in the same group as 1. So, there are two possible groups left.
- For the number 9, since 2x ≤ 10, it can't be in the same group as 2 or 3. So, there is only one possible group left.
- For the number 10, since 2x ≤ 10, it can't be in the same group as 1, 2, or 3. So, there are no options left.
Based on the above analysis, we have the following possibilities for placing the numbers:
1. Group 1: {1, 2, 4, 8} | Group 2: {3, 6} | Group 3: {5, 7, 9, 10}
2. Group 1: {1, 2, 4, 8} | Group 2: {3, 6, 9} | Group 3: {5, 7, 10}
3. Group 1: {1, 2, 4, 8} | Group 2: {3, 6, 9, 10} | Group 3: {5, 7}
4. Group 1: {1, 2, 4} | Group 2: {3, 6} | Group 3: {5, 7, 9, 10}
5. Group 1: {1, 2, 4} | Group 2: {3, 6, 9} | Group 3: {5, 7, 10}
6. Group 1: {1, 2, 4} | Group 2: {3, 6, 9, 10} | Group 3: {5, 7}
7. Group 1: {1, 2} | Group 2: {3, 6} | Group 3: {5, 7, 9, 10}
8. Group 1: {1, 2} | Group 2: {3, 6, 9} | Group 3: {5, 7, 10}
9. Group 1: {1, 2} | Group 2: {3, 6, 9, 10} | Group 3: {5, 7}
So, there are a total of 9 different ways to create the groups.