<p>Rekha married Shivram and had 4 sons. Varsha married Ajoy and had 4 sons. Both the couples had divorce and after that Shivram married Varsha while Ajoy married Rekha. They too had 3 sons each from their wedlocks. How many selections of 8 children can be made from the 14 children so that each of them have equal number of sons in the selection?</p>
Step-by-Step Solution
Key Concept: We have 4 distinct groups of children (each parent couple's biological children), and we must select 8 children such that from each group, we pick equal numbers. This is a constraint-based selection problem where we need to distribute 8 selections equally among 4 groups.
<p><strong>Step 1: Identify the four groups</strong></p><p>• Group A (Rekha-Shivram): 4 sons</p><p>• Group B (Varsha-Ajoy): 4 sons</p><p>• Group C (Shivram-Varsha): 3 sons</p><p>• Group D (Ajoy-Rekha): 3 sons</p><p>Total: 14 children</p><p><strong>Step 2: Determine equal distribution constraint</strong></p><p>We need to select 8 children with equal number from each group. If we select k children from each of 4 groups: 4k = 8, so k = 2</p><p><strong>Step 3: Check feasibility</strong></p><p>We need to select exactly 2 children from each group.</p><p>• From Group A (4 sons): C(4,2) = 6 ways</p><p>• From Group B (4 sons): C(4,2) = 6 ways</p><p>• From Group C (3 sons): C(3,2) = 3 ways</p><p>• From Group D (3 sons): C(3,2) = 3 ways</p><p><strong>Step 4: Apply multiplication principle</strong></p><p>Total selections = C(4,2) × C(4,2) × C(3,2) × C(3,2)</p><p>= 6 × 6 × 3 × 3 = 36 × 9 = 324</p><p><strong>∴ Answer: 324</strong></p>
Correct Answer: 324