<p>Find the number of ways in which four distinct balls can be kept into two identical boxes so that no box remains empty.</p>
Step-by-Step Solution
Key Concept: With identical boxes, arrangements that differ only by swapping boxes are the same. Partition the 4 balls into two non-empty groups, then divide by 2 to account for box indistinguishability.
<p><strong>Step 1:</strong> Find total ways to distribute 4 distinct balls into 2 distinguishable boxes with no empty box.</p><p>Total distributions = 2⁴ = 16. Subtract cases where at least one box is empty: 16 - 2 = 14 ways.</p><p><strong>Step 2:</strong> Since the boxes are identical, arrangements differing only by swapping boxes are the same. Each valid partition is counted twice in our calculation.</p><p>Number of ways = 14 ÷ 2 = 7</p><p><strong>Verification:</strong> Partitions are: {1,3}, {2,2}, {3,1} but {1,3} and {3,1} are same when boxes are identical. Actual partitions: (1-3 split): C(4,1)=4 ways, (2-2 split): C(4,2)/2=3 ways. Total = 4 + 3 = 7.</p><p>∴ <strong>Answer: 7</strong></p>
Correct Answer: 7