<p>A bag contains four one-rupee coins, two twenty-five paisa coins, and five ten-paisa coins. In how many ways can an amount, not less than ₹1 be taken out from the bag? (Consider coins of the same denominations to be identical.)</p>
Step-by-Step Solution
Key Concept: The total number of ways to select coins is the product of independent choices for each denomination, then subtract 1 for the empty selection and subtract cases giving amounts less than ₹1.
<p><strong>Step 1:</strong> For each denomination, find the number of ways to select coins.</p><p>• One-rupee coins (4 coins): Can select 0, 1, 2, 3, or 4 coins → 5 ways</p><p>• Twenty-five paisa coins (2 coins): Can select 0, 1, or 2 coins → 3 ways</p><p>• Ten-paisa coins (5 coins): Can select 0, 1, 2, 3, 4, or 5 coins → 6 ways</p><p><strong>Step 2:</strong> Total selections = 5 × 3 × 6 = 90 ways (includes selecting no coins).</p><p><strong>Step 3:</strong> Subtract 1 for the empty selection (₹0): 90 - 1 = 89 ways.</p><p><strong>Step 4:</strong> Now subtract selections giving amounts less than ₹1 (i.e., less than 100 paisa):</p><p>• Zero one-rupee coins AND total from other denominations < 100 paisa:</p><p> - With 0 rupees: can use (0 to 2 twenty-five paisa) and (0 to 5 ten-paisa)</p><p> - Maximum from 25-paisa: 2 × 25 = 50 paisa</p><p> - From 10-paisa: 0, 1, 2, 3 coins give 0, 10, 20, 30 paisa (4 ways)</p><p> - From 25-paisa: 0, 1, 2 coins (3 ways)</p><p> - Valid combinations with 0 rupees: 3 × 4 = 12 ways (all ≤ 95 paisa)</p><p> - Subtract case (0, 0, 0): 12 - 1 = 11 ways giving amounts < ₹1</p><p><strong>Step 5:</strong> Valid selections = 89 - 11 = 78</p><p>∴ Answer: 78 ways</p>
Correct Answer: C