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

(a) 16

(b) 15

(c) 14

(d) 10

Correct Answer: (a) 16

#### Question from UPSC Prelims 2020 CSAT Paper

**Explanation :**

## Different sums that can be formed with different denominations

The number of different sums that can be formed with the denominations Rs. 50, Rs. 100, Rs. 200, Rs. 500 and Rs. 2,000 taking at least three denominations at a time is 16.

Here’s how we can calculate it: There are five denominations in total and we need to choose at least three of them to form a sum. We can use the formula for combinations to calculate the number of ways to choose k items from n items: C(n,k) = n! / (k!(n-k)!), where n is the total number of items and k is the number of items being chosen.

Using this formula, we can calculate the number of ways to choose three denominations from five:** C(5,3) = 5! / (3!(5-3)!) = 10**.

Similarly, we can calculate the number of ways to choose four denominations from five: **C(5,4) = 5! / (4!(5-4)!) = 5**.

And finally, there is only one way to choose all five denominations.

Adding up all these possibilities gives us a total of 10 + 5 + 1 = 16 different sums that can be formed using at least three denominations.