<p>In how many ways can ₹16 be divided into 4 persons when none of them gets less than ₹3?</p>
Step-by-Step Solution
Key Concept: Transform the constrained problem into an unconstrained one by first giving each person ₹3 (satisfying the minimum), then distribute the remaining ₹4 freely among 4 persons using stars and bars.
<p><strong>Step 1:</strong> Apply the minimum constraint. Give each of the 4 persons ₹3 initially.</p><p>Amount distributed = 4 × 3 = ₹12</p><p>Remaining amount = 16 − 12 = ₹4</p><p><strong>Step 2:</strong> Now distribute the remaining ₹4 among 4 persons with no restrictions using stars and bars formula: C(n+k−1, k−1)</p><p>Here, n = 4 (remaining amount), k = 4 (persons)</p><p>Number of ways = C(4+4−1, 4−1) = C(7, 3)</p><p><strong>Step 3:</strong> Calculate C(7, 3)</p><p>C(7, 3) = 7!/(3! × 4!) = (7 × 6 × 5)/(3 × 2 × 1) = 210/6 = 35</p><p><strong>∴ Answer: 35</strong></p>
Correct Answer: 35