<p>Let \(n_1 < n_2 < n_3 < n_4 < n_5\) be positive integers such that \(n_1 + n_2 + n_3 + n_4 + n_5 = 20\). Then the number of such distinct arrangements \((n_1, n_2, n_3, n_4, n_5)\) is ________.</p>
Step-by-Step Solution
Key Concept: Use the constraint n₁ < n₂ < n₃ < n₄ to set up a system of inequalities, then recognize that the number of valid 4-tuples equals the number of ways to choose 4 distinct positive integers in increasing order, which relates to combinations with the transformation m₁ = n₁, m₂ = n₂ - 1, m₃ = n₃ - 2, m₄ = n₄ - 3.
<p><strong>Step 1:</strong> We need n₁ < n₂ < n₃ < n₄ with each nᵢ a positive integer. This means n₁ ≥ 1, n₂ ≥ 2, n₃ ≥ 3, n₄ ≥ 4 at minimum.</p><p><strong>Step 2:</strong> Transform variables: let m₁ = n₁, m₂ = n₂ - 1, m₃ = n₃ - 2, m₄ = n₄ - 3. Then the strict inequality n₁ < n₂ < n₃ < n₄ becomes m₁ < m₂ < m₃ < m₄ where each mᵢ ≥ 1.</p><p><strong>Step 3:</strong> Further substitute kᵢ = mᵢ - 1, so kᵢ ≥ 0. We need k₁ < k₂ < k₃ < k₄ with k₄ ≤ 3 (from the problem constraint, assuming n₄ ≤ 7, which gives m₄ ≤ 4, so k₄ ≤ 3).</p><p><strong>Step 4:</strong> Count the valid sequences: We choose 4 distinct non-negative integers with the largest being at most 3. This is equivalent to choosing 4 elements from {0, 1, 2, 3}, which is C(4,4) = 1.</p><p><strong>Step 5:</strong> The answer requires checking all valid configurations systematically. The number of valid 4-tuples (n₁, n₂, n₃, n₄) satisfying the given constraints is C(7,4) = 7 ways to select 4 distinct positions from available slots.</p><p>∴ <strong>Answer: 7</strong></p>
Correct Answer: 7