How Many Different Sums Can Be Formed with Various Coin Denominations?

When dealing with different coin denominations, such as rupee, paise, and paisa, it can be interesting to explore how many unique sums can be formed. In this article, we will explore the process of determining the number of different sums that can be created using various coin denominations. We will utilize both the dynamic programming approach and a combinatorial method to solve this problem.

Step 1: Convert Denominations to the Smallest Unit (Paise)

To simplify the process, we need to convert all denominations into the smallest unit, paise. Here are the conversions:

5 rupees 500 paise 1 rupee 100 paise 50 paise 50 paise 35 paise 35 paise 10 paise 10 paise 3 paise 3 paise 2 paise 2 paise 1 paisa 1 paise

Step 2: List of Coin Denominations in Paise

Now that we have all denominations in paise, we can list them as follows:

500 paise 100 paise 50 paise 35 paise 10 paise 3 paise 2 paise 1 paise

Step 3: Finding Possible Sums

There are two main methods to find the number of different sums that can be formed with these coins: dynamic programming and combinatorial approach.

Dynamic Programming Approach

The dynamic programming (DP) approach involves using a Boolean array to track whether a sum can be formed with the available coins. Here are the steps:

Initialize a Boolean array dp[n 1] where dp[i] is True if the sum i paise can be formed, and False otherwise. dp[0] True Iterate through each coin. For each coin, update the dp array for possible sums from the coin value to the maximum possible sum.

Combinatorial Approach

The combinatorial approach involves considering each coin and deciding whether to include it or not in the sum. This approach is more intuitive but can be computationally intensive for large denominations.

Step 4: Calculate Maximum Possible Sum

The maximum possible sum we can form using all the coins is the sum of their values:

500 100 50 35 10 3 2 1 701 paise

Step 5: Count the Different Sums

After processing all the coins, count how many entries in the dp array are True from 0 to 701. This count represents the number of different sums that can be formed.

Conclusion

The number of different sums that can be formed is the count of True values in the dp array from 0 to 701. This approach can be implemented in code or calculated manually for a smaller set of denominations.

Key Points to Remember

Dynamic Programming: A method to track possible sums using a Boolean array. Combinatorial Approach: Deciding to include or exclude each coin. Dynamic Array Update: Updating the dp array for each coin value.

If you need help with the specific count of different sums, feel free to reach out for assistance.