Permutations & Combinations Question - None

Question

Find the number of ordered pairs $(x, y)$ if $x, y \in \{0, 1, 2, 3, \ldots, 10\}$ and if $|x - y| > 5$.

Accepted Answer

Step 1: Recognize that |x - y| > 5 splits into two cases: x - y > 5 OR y - x > 5.

Step 2: For case x - y > 5 (i.e., x ≥ y + 6):

• If y = 0: x ∈ {6,7,8,9,10} → 5 pairs

• If y = 1: x ∈ {7,8,9,10} → 4 pairs

• If y = 2: x ∈ {8,9,10} → 3 pairs

• If y = 3: x ∈ {9,10} → 2 pairs

• If y = 4: x ∈ {10} → 1 pair

• If y ≥ 5: no valid x

Subtotal: 5 + 4 + 3 + 2 + 1 = 15 pairs

Step 3: By symmetry, case y - x > 5 (i.e., y ≥ x + 6) also yields 15 pairs.

Step 4: No overlap exists since |x - y| > 5 cannot equal both x - y > 5 and y - x > 5 simultaneously.

∴ Answer: 15 + 15 = 30

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