To solve the problem of finding the number of non-empty subsets of the set {1, 2, 3, ..., 12} such that the sum of the largest and smallest element is 13, we can follow these steps:
### Step 1: Identify pairs of smallest and largest elements
We need to find pairs (smallest, largest) such that their sum equals 13. The possible pairs from the set {1, 2, ..., 12} are:
- (1, 12)
- (2, 11)
- (3, 10)
- (4, 9)
- (5, 8)
- (6, 7)
### Step 2: Count elements between the smallest and largest
For each of these pairs, we need to consider the elements that can be chosen from the numbers between the smallest and largest elements.
1. For the pair (1, 12):
- Elements between: {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} (10 elements)
- Number of subsets: \(2^{10}\)
2. For the pair (2, 11):
- Elements between: {3, 4, 5, 6, 7, 8, 9, 10} (8 elements)
- Number of subsets: \(2^{8}\)
3. For the pair (3, 10):
- Elements between: {4, 5, 6, 7, 8, 9} (6 elements)
- Number of subsets: \(2^{6}\)
4. For the pair (4, 9):
- Elements between: {5, 6, 7, 8} (4 elements)
- Number of subsets: \(2^{4}\)
5. For the pair (5, 8):
- Elements between: {6, 7} (2 elements)
- Number of subsets: \(2^{2}\)
6. For the pair (6, 7):
- No elements between (0 elements)
- Number of subsets: \(2^{0} = 1\) (the subset {6, 7} itself)
### Step 3: Calculate total number of subsets
Now, we can sum the number of subsets for each pair:
- For (1, 12): \(2^{10} = 1024\)
- For (2, 11): \(2^{8} = 256\)
- For (3, 10): \(2^{6} = 64\)
- For (4, 9): \(2^{4} = 16\)
- For (5, 8): \(2^{2} = 4\)
- For (6, 7): \(2^{0} = 1\)
Total number of non-empty subsets = \(1024 + 256 + 64 + 16 + 4 + 1 = 1365\)
### Final Answer
The total number of non-empty subsets of {1, 2, 3, ..., 12} such that the sum of the largest and smallest element is 13 is **1365**.
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