How many different sums can be formed with the denominations ₹50, ₹100, ₹200,₹500 and ₹2,000 taking at least three denominations at a time?

The correct answer is Option 16.

To find the total number of different sums that can be formed by using at least three denominations at a time, we can analyze the given denominations and use combinatorics.

1. Identify the given denominations:
We have 5 distinct currency denominations:
₹50, ₹100, ₹200, ₹500, and ₹2,000.
Let $n=5$ be the total number of available denominations.

2. Key Observation:
Since all 5 denominations are distinct and each sum formed by taking a distinct subset of these notes results in a unique total value (because no combination of smaller notes equals a single larger note in a way that creates overlap among these specific subset sums), every unique combination of denominations will yield a unique sum.

3. Calculate the combinations for taking "at least three" denominations at a time:
"At least three" means we can choose 3, 4, or all 5 denominations.

The number of ways to choose $k$ denominations out of $n$ is given by the combination formula:
$C(n,k)=\frac{n!}{k!(n-k)!}$

Now, let us calculate each case:

Case 1: Taking exactly 3 denominations at a time ($k=3$)
$C(5,3)=\frac{5\times 4\times 3}{3\times 2\times 1}=10$

Case 2: Taking Service/exactly 4 denominations at a time ($k=4$)
$C(5,4)=\frac{5\times 4\times 3\times 2}{4\times 3\times 2\times 1}=5$

Case 3: Taking all 5 denominations at a time ($k=5$)
$C(5,5)=1$

4. Calculate the Total Number of Sums:
Total sums = $C(5,3)+C(5,4)+C(5,5)$
Total sums = $10+5+1=16$

Therefore, a total of 16 different sums can be formed.