Let P be the partial order defined on the set \1,2,3,4\ as follows P = \(x,x) x \1,2,3,4\\ \(1,2), (3,2), (3,4)\ The number of total orders on \1,2,3,4\ that contain P is …

Official answer: 5. Explanation: A total order containing a partial order P is also known as a linear extension of P. The given partial order P imposes the following strict precedence relations between distinct elements: - 1 < 2 - 3 < 2 - 3 < 4 We need to find the number of permutations (x1, x2, x3, x4) of the set \1, 2, 3, 4\ that satisfy these three constraints. The minimal elements of the poset (elements with no predecessors) are 1 and 3. Case 1: 1 appears first. - The permutation begins with 1: (1, \, \, \). - The remaining elements to arrange are \2, 3, 4\. - Since 3 < 2 and 3 < 4, 3 must precede both 2 and 4. - Thus, 3 must be in the second position: (1, 3, \, \). - The remaining elements \2, 4\ have no relative precedence between them, giving 2 valid permutations: 1. 1 < 3 < 2 < 4 2. 1 < 3 < 4 < 2 Case 2: 3 appears first. - The permutation begins with 3: (3, \, \, \). - The remaining elements to arrange are \1, 2, 4\. - The only remaining constraint is 1 < 2, while 4 can be placed in any position relative to 1 and 2. - The number of ways to arrange \1, 2, 4\ such that 1 < 2 is: 3!2! = 3 - The 3 valid permutations are: 1. 3 < 4 < 1 < 2 2. 3 < 1 < 4 < 2 3. 3 < 1 < 2 < 4 Adding both cases together: Total number of linear extensions = 2 + 3 = 5 Thus, the number of total orders on \1,2,3,4\ that contain P is 5.

Engineering Mathematics · Discrete Mathematics

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GATE CSE 2024 Set 2 Q34 · NAT · 1 mark

Let $P$ be the partial order defined on the set $\{1,2,3,4\}$ as follows $P = \{(x,x) \mid x \in \{1,2,3,4\}\} \cup \{(1,2), (3,2), (3,4)\}$ The number of total orders on $\{1,2,3,4\}$ that contain $P$ is _________

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Discrete Mathematics

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In GATE Computer Science, Discrete Mathematics is heavily tested across mathematical logic, set theory, relations & partial orders/lattices, group theory, combinatorics, and graph theory. Questions evaluate formal reasoning, counterexample construction, and structural counting principles. Over recent years, question formats have transitioned from standard single-choice definitions to multi-select properties (MSQs) and exact integer/combinatorial numerical answers (NATs).

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Engineering Mathematics · Discrete Mathematics

Official IIT answer key · IISc Bangalore · Audited Aug 2026

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GATE CSE 2024 Set 2 Q34 · NAT · 1 mark

Let $P$ be the partial order defined on the set $\{1,2,3,4\}$ as follows $P = \{(x,x) \mid x \in \{1,2,3,4\}\} \cup \{(1,2), (3,2), (3,4)\}$ The number of total orders on $\{1,2,3,4\}$ that contain $P$ is _________

Numerical answer

Enter a value to check

Key concept

Discrete Mathematics

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Examiner trap & misconception

Topic notes

How GATE tests Discrete Mathematics

Full Discrete Mathematics guide →

More on Discrete Mathematics

All 50 questions on Discrete Mathematics →

Independent practice explanation verified by the GateAI team. GATE is conducted by the IITs; question text follows the official paper.

Let $P$ be the partial order defined on the set $\{1,2,3,4\}$ as follows $P = \{(x,x) \mid x \in \{1,2,3,4\}\} \cup \{(1,2), (3,2), (3,4)\}$ The number of total orders on $\{1,2,3,4\}$ that contain $P$ is _________

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Let $P$ be the partial order defined on the set $\{1,2,3,4\}$ as follows $P = \{(x,x) \mid x \in \{1,2,3,4\}\} \cup \{(1,2), (3,2), (3,4)\}$ The number of total orders on $\{1,2,3,4\}$ that contain $P$ is _________

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