<p>A bag contains 4 one-rupee coins, 2 twenty-five-paisa coins and 5 ten-paisa coins. In how many ways can an amount not less than Re 1 be taken out from the bag? Consider coins of the same denomination to be identical.</p>
Step-by-Step Solution
Key Concept: Since coins of the same denomination are identical, we only need to count the number of ways to select quantities of each type (not individual coins). The problem reduces to counting non-negative integer solutions with constraints, excluding the case where we take nothing.
<p><strong>Step 1: Identify choices for each denomination</strong></p><p>Since coins of the same denomination are identical, we select quantities:</p><ul><li>One-rupee coins (₹1 each): Choose 0, 1, 2, 3, or 4 → <strong>5 ways</strong></li><li>Twenty-five paisa coins (₹0.25 each): Choose 0, 1, or 2 → <strong>3 ways</strong></li><li>Ten-paisa coins (₹0.10 each): Choose 0, 1, 2, 3, 4, or 5 → <strong>6 ways</strong></li></ul><p><strong>Step 2: Calculate total combinations</strong></p><p>By multiplication principle: 5 × 3 × 6 = <strong>90 ways</strong> (including taking nothing)</p><p><strong>Step 3: Exclude the null case</strong></p><p>We must exclude selecting (0, 0, 0) which gives amount = Re 0, as we need "not less than Re 1"</p><p>Valid selections = 90 - 1 = <strong>89</strong></p><p>∴ Answer: <strong>89 ways</strong></p>
Correct Answer: 89