<p>Find the number of ways in which 13 identical apples can be distributed among 3 persons so that no two persons receive equal number of apples and each can receive any number of apples.</p>
Step-by-Step Solution
Key Concept: Use stars and bars with the constraint that all three persons receive distinct numbers of apples. The key is to count ordered triples (a,b,c) where a+b+c=13 and a,b,c are all different non-negative integers.
<p><strong>Step 1:</strong> We need to find non-negative integer solutions to a+b+c=13 where a≠b, b≠c, and a≠c.</p><p><strong>Step 2:</strong> Use complementary counting. Total unrestricted solutions = C(13+3-1, 3-1) = C(15,2) = 105.</p><p><strong>Step 3:</strong> Subtract cases where at least two are equal:</p><ul><li>If a=b: Then 2a+c=13, so c=13-2a. For distinct values, a≠c means a≠13-2a, so 3a≠13. Since a∈{0,1,...,6}, we get 7 solutions (a can be 0,1,2,3,4,5,6).</li><li>Similarly for b=c and a=c: 7 solutions each.</li><li>If a=b=c: Then 3a=13, which has no integer solution.</li></ul><p><strong>Step 4:</strong> By inclusion-exclusion: Invalid cases = 7+7+7-0 = 21</p><p><strong>Step 5:</strong> Valid distributions = 105-21 = 84</p><p>∴ Answer: 84</p>
Correct Answer: 84