Permutations & Combinations
Counting Solutions to Equations
Grade 11
Question:
<p>In how many different ways can a sum of Rs 20 be paid in one-rupee coins, 50-paisa coins and 25-paisa coins if each variety of coins is available in unlimited number?</p>
Step-by-Step Solution
Key Concept: Convert all coins to a common unit (paisa) and find non-negative integer solutions to 100a + 50b + 25c = 2000, then count valid combinations by systematically varying each variable.
<p><strong>Step 1:</strong> Convert to paisa: 20 Rs = 2000 paisa. Let a = number of 100-paisa (1 Re) coins, b = number of 50-paisa coins, c = number of 25-paisa coins.</p><p><strong>Step 2:</strong> We need: 100a + 50b + 25c = 2000. Divide by 25: 4a + 2b + c = 80, so c = 80 - 4a - 2b.</p><p><strong>Step 3:</strong> For valid solutions: c ≥ 0 ⟹ 80 - 4a - 2b ≥ 0 ⟹ 2a + b ≤ 40.</p><p><strong>Step 4:</strong> For each value of a (where 0 ≤ a ≤ 20), b can range from 0 to (40 - 2a). Number of valid b values = 40 - 2a + 1 = 41 - 2a.</p><p><strong>Step 5:</strong> Total ways = Σ(41 - 2a) for a = 0 to 20<br>= (41 + 39 + 37 + ... + 3 + 1)<br>= sum of first 21 odd numbers<br>= 21²</p><p>∴ Answer: <strong>441</strong></p>
Correct Answer: 441