A person X 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?

Let x be the number of ₹2 coins and y be the number of ₹5 coins used. The total payment can be expressed by the linear equation:

2x + 5y = 56

Since 56 and 2x are even numbers, 5y must also be even, which implies y must be an even integer. We are given the constraints 0 ≤ x ≤ 25 and 0 ≤ y ≤ 9. We can test the possible even values for y:

• If y = 0: 2x = 56 → x = 28 (Invalid, as only 25 coins are available)
• If y = 2: 2x = 46 → x = 23 (Valid)
• If y = 4: 2x = 36 → x = 18 (Valid)
• If y = 6: 2x = 26 → x = 13 (Valid)
• If y = 8: 2x = 16 → x = 8 (Valid)

For y ≥ 10, the value exceeds the available nine ₹5 coins. Therefore, there are exactly 4 different ways to make the payment.