<p>Number of non-empty subsets of \(\{1, 2, 3, \ldots, 12\}\) having the property that sum of the largest and smallest element is 13.</p>
Step-by-Step Solution
Key Concept: If a subset has smallest element $a$ and largest element $b$ with $a + b = 13$, then $a$ and $b$ are fixed, and we can freely choose any subset of the elements strictly between them. The pairs $(a,b)$ satisfying $a + b = 13$ with $a < b$ are: $(1,12), (2,11), (3,10), (4,9), (5,8), (6,7)$.
<p><strong>Step 1:</strong> Identify the constraint. We need subsets where min + max = 13, with min < max.</p><p><strong>Step 2:</strong> Find all valid pairs $(a, b)$ where $a + b = 13$ and $1 \leq a < b \leq 12$:</p><p>$(1,12), (2,11), (3,10), (4,9), (5,8), (6,7)$ — that's 6 pairs.</p><p><strong>Step 3:</strong> For each pair $(a, b)$, both $a$ and $b$ must be in the subset. The elements strictly between $a$ and $b$ are: $\{a+1, a+2, \ldots, b-1\}$, which contains $b - a - 1$ elements.</p><p><strong>Step 4:</strong> Each of these middle elements can either be included or excluded, giving $2^{b-a-1}$ choices.</p><p><strong>Step 5:</strong> Calculate for each pair:</p><ul><li>$(1,12)$: $2^{12-1-1} = 2^{10} = 1024$</li><li>$(2,11)$: $2^{11-2-1} = 2^{8} = 256$</li><li>$(3,10)$: $2^{10-3-1} = 2^{6} = 64$</li><li>$(4,9)$: $2^{9-4-1} = 2^{4} = 16$</li><li>$(5,8)$: $2^{8-5-1} = 2^{2} = 4$</li><li>$(6,7)$: $2^{7-6-1} = 2^{0} = 1$</li></ul><p><strong>Step 6:</strong> Total = $1024 + 256 + 64 + 16 + 4 + 1 = 1365$</p><p>∴ Answer: <strong>1365</strong></p>
Correct Answer: 1365