A person has to make a payment of ₹56. He has twenty-five ₹2 coins and nine ₹5 coins. In how many different ways can he make this payment?
4
Let x be the number of ₹2 coins used and y the number of ₹5 coins used, with \(2x+5y=56\), \(0\le x\le25\), \(0\le y\le9\).
Since 56 and 2x are even, 5y must be even, so y must be even.
y = 0: x = 28 (exceeds 25, rejected). y = 2: x = 23 (valid). y = 4: x = 18 (valid). y = 6: x = 13 (valid). y = 8: x = 8 (valid). y = 10 exceeds the 9 coins available.
So the valid combinations are y = 2, 4, 6, 8 — 4 different ways.
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