Permutations & Combinations
Counting with Ordering Conditions
Grade 11
Question:
<p>Consider three boxes, each containing 10 balls labelled 1, 2, ..., 10. Suppose one ball is randomly drawn from each of the boxes. Denote by \(n_i\) the label of the ball drawn from the \(i^{\text{th}}\) box, \((i = 1, 2, 3)\). Then, the number of ways in which the balls can be chosen such that \(n_1 < n_2 < n_3\) is ______.</p>
Step-by-Step Solution
Key Concept: The constraint n₁ < n₂ < n₃ means we need strictly increasing labels across the three draws. This is equivalent to choosing 3 distinct numbers from {1,2,...,10} and arranging them in increasing order—which happens in exactly one way for any such selection.
<p><strong>Step 1:</strong> Recognize the problem structure. We draw one ball from each of three boxes, and need the labels to satisfy n₁ < n₂ < n₃.</p><p><strong>Step 2:</strong> For any choice of three distinct labels from {1, 2, ..., 10}, there is exactly ONE way to arrange them in strictly increasing order. Therefore, choosing balls with n₁ < n₂ < n₃ is equivalent to choosing an unordered set of 3 distinct numbers from 10 numbers.</p><p><strong>Step 3:</strong> The number of ways to choose 3 distinct numbers from 10 is C(10,3) = 10!/(3!×7!) = (10 × 9 × 8)/(3 × 2 × 1) = 720/6 = 120.</p><p><strong>Verification:</strong> Total unrestricted outcomes = 10³ = 1000. The constraint n₁ < n₂ < n₃ selects exactly those outcomes where labels are distinct and in increasing order, giving us C(10,3) = 120.</p><p>∴ Answer: <strong>120</strong></p>
Correct Answer: 120