Permutations & Combinations
Distribution of identical objects
Grade 11
Question:
<p><strong>Statement 1:</strong> The number of ways of distributing 10 identical balls in 4 distinct boxes such that no box is empty is \(^9C_3\).<br><strong>Statement 2:</strong> The number of ways of choosing any 3 places from 9 different places is \(^9C_3\).</p><p>(1) Statement 1 is false, statement 2 is true.</p>
<p>(1) Statement 1 is false, statement 2 is true.</p>
<p>(2) Statement 1 is true, statement 2 is true; statement 2 is a correct explanation for statement 1.</p>
<p>(3) Statement 1 is true, statement 2 is true; statement 2 is not a correct explanation for statement 1.</p>
<p>(4) Statement 1 is true, statement 2 is false.</p>
Step-by-Step Solution
Key Concept: Distributing n identical objects into k distinct boxes with no empty box is equivalent to arranging (n-1) dividers among (n-1) positions, which equals C(n-1, k-1). For 10 balls in 4 boxes: C(9,3) = 84. Statement 2 is simply the definition of combinations, which is also C(9,3) = 84. Both statements are numerically identical AND conceptually valid.
<p><strong>Step 1:</strong> Verify Statement 1 using Stars and Bars formula.</p><p>For distributing n identical objects into k distinct boxes with NO empty box: Place 1 ball in each box first (uses 10 balls minimally), then distribute remaining (10-4)=6 balls freely among 4 boxes.</p><p>This is equivalent to: C(6+4-1, 4-1) = C(9,3) ✓</p><p>Alternatively: Think of 10 balls as ●●●●●●●●●● and 3 dividers as ||. We need to choose 3 positions from 9 gaps between balls for dividers: C(9,3) ✓</p><p><strong>Step 2:</strong> Verify Statement 2.</p><p>Choosing any 3 places from 9 different places = C(9,3) ✓</p><p><strong>Step 3:</strong> Evaluate truth values.</p><p>Statement 1: TRUE (equals C(9,3) by correct formula)</p><p>Statement 2: TRUE (direct combination definition)</p><p><strong>Step 4:</strong> Check given options.</p><p>Option (1) states: 'Statement 1 is false, statement 2 is true' - This is INCORRECT.</p><p>Both statements are TRUE, so the correct answer must be option (2): 'Both statements are true.'</p><p>∴ Answer: 2</p>
Correct Answer: 2