Calculating different sums using denominations
The given denominations are 50, 100, 200, 500, and 2000. We need to find out how many different sums can be formed by taking 3 denominations at a time.

Step 1: Determine all possible combinations
To find all possible combinations, we need to consider all unique combinations of 3 denominations out of the 5 given denominations. This can be calculated using the combination formula:
nCr = n! / r!(n-r)!
For our case, n = 5 (number of denominations) and r = 3 (taking 3 denominations at a time).
5C3 = 5! / 3!(5-3)! = 5! / 3!2! = 10
Therefore, there are a total of 10 possible combinations of 3 denominations that can be taken at a time.

Step 2: Calculate the sums for each combination
Now, we need to calculate the sum for each combination. This can be done by finding the sum of the 3 denominations selected in each combination.
For example, if we select denominations 50, 100, and 200, the sum would be 50 + 100 + 200 = 350.

Step 3: Count the unique sums
After calculating the sums for each combination, we need to count the number of unique sums that can be formed. This can be done by listing out all the sums and removing any duplicates.

Step 4: Final count of different sums
By following the above steps, we can determine the total number of different sums that can be formed using denominations of 50, 100, 200, 500, and 2000, taking 3 at a time.