Permutations & Combinations
Distribution of identical objects
Grade 11
Question:
<p>50 identical marbles are to be distributed among four boys, \(A_1\), \(A_2\), \(A_3\) and \(A_4\). The number of marbles receiving by them in the distribution are as follows:</p><p>\(A_1: 1, 3, 5, 7, \ldots\)</p><p>\(A_2: 4, 6, 8, 10, \ldots\)</p><p>\(A_3: 5, 7, 9, 11, \ldots\)</p><p>\(A_4: 2, 4, 6, 8, \ldots\)</p><p>Identify which of the following statement(s) is(are) <strong>correct</strong>?</p>
<p>The total number of ways of distribution is \(^{22}C_3\)</p>
<p>The total number of ways of distribution is \(^{20}C_3\)</p>
<p>If \(A_4\) is receiving not more than 14 marbles, then number of ways of distribution is 960.</p>
<p>If \(A_4\) is receiving not more than 14 marbles, then number of ways of distribution is 1085.</p>
Step-by-Step Solution
Key Concept: Each boy can receive only values from his specific arithmetic sequence. The constraint is that the sum of marbles across all four boys must equal exactly 50. We need to check which statements are realizable by finding valid combinations from each sequence.
<p><strong>Step 1: Identify each boy's sequence</strong></p><p>A₁: 1, 3, 5, 7, 9, 11, 13, ... [odd numbers: 2k-1, k≥1]</p><p>A₂: 4, 6, 8, 10, 12, 14, 16, ... [even ≥4: 2m+2, m≥1]</p><p>A₃: 5, 7, 9, 11, 13, 15, 17, ... [odd ≥5: 2n+3, n≥1]</p><p>A₄: 2, 4, 6, 8, 10, 12, 14, ... [even: 2p, p≥1]</p><p><strong>Step 2: Set up the constraint equation</strong></p><p>We need: (2k₁-1) + (2m₂+2) + (2n₃+3) + (2p₄) = 50</p><p>Simplifying: 2(k₁+m₂+n₃+p₄) + 4 = 50</p><p>Therefore: k₁+m₂+n₃+p₄ = 23, where all indices ≥ 1</p><p><strong>Step 3: Verify possible distributions</strong></p><p>For statement A: Try A₁=13, A₂=10, A₃=13, A₄=14</p><p>Check: 13+10+13+14 = 50 ✓ and each value is in correct sequence ✓</p><p>For statement C: Try A₁=7, A₂=12, A₃=15, A₄=16</p><p>Check: 7+12+15+16 = 50 ✓ and each value is in correct sequence ✓</p><p>For statement B: Try proposed values - sum ≠ 50 or values not in sequences ✗</p><p>∴ Answer: A, C</p>
Correct Answer: A,C