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'll break it down into smaller steps.
Step 1: Determine the largest integer that can be in the first group.
Since the groups contain at least one number, the largest integer that can be in the first group is 9. If we put 10 in the first group, there will be no integer that satisfies the property 2x≤10.
Step 2: Determine the largest integer that can be in the second group.
Since the second group must satisfy the property that if x is in the group, then 2x is also in the group, the largest integer that can be in the second group is one that is less than or equal to half of the largest integer in the first group. In this case, since the largest integer in the first group is 9, the largest integer that can be in the second group is 4.
Step 3: Determine the largest integer that can be in the third group.
Similarly, the third group must satisfy the property that if x is in the group, then 2x is also in the group. The largest integer that can be in the third group is one that is less than or equal to half of the largest integer in the second group. In this case, since the largest integer in the second group is 4, the largest integer that can be in the third group is 2.
Step 4: Determine the number of ways to create the groups.
Now that we have determined the maximum integers that can be in each group, we can count the number of ways to arrange them. We have three groups, and for each group, we have multiple options. The possible numbers for the first group are 1, 2, 3, 4, 5, 6, 7, 8, or 9. The possible numbers for the second group are 1, 2, 3, or 4. The possible numbers for the third group are 1 or 2.
To find the total number of ways, we multiply the number of options for each group. In this case, the number of ways to create the groups is 9 * 4 * 2 = 72.
Therefore, there are 72 different ways to create the groups.