In how many ways can X give Rs.500 to Y using only Rs.100 and Rs.20 notes, with the condition that she has only 16 notes of Rs.20, and being asked to use the notes of both the denominations?

To solve this problem, we need to consider the different combinations of Rs.100 and Rs.20 notes that can add up to Rs.500.

1. First, let's consider the Rs.100 notes. The number of Rs.100 notes can be 0, 1, 2, 3, 4, or 5 because we cannot exceed Rs.500.

2. For each number of Rs.100 notes, we need to calculate the remaining amount that needs to be made up with Rs.20 notes.

3. The remaining amount must be divisible by 20, and we cannot use more than 16 Rs.20 notes.

Let's calculate:

• If X uses 0 Rs.100 notes, she needs to give 25 Rs.20 notes, but she only has 16. So, this is not possible.
• If X uses 1 Rs.100 note, she needs to give 20 Rs.20 notes, but she only has 16. So, this is not possible.
• If X uses 2 Rs.100 notes, she needs to give 15 Rs.20 notes, which is possible.
• If X uses 3 Rs.100 notes, she needs to give 10 Rs.20 notes, which is possible.
• If X uses 4 Rs.100 notes, she needs to give 5 Rs.20 notes, which is possible.
• If X uses 5 Rs.100 notes, she needs to give 0 Rs.20 notes, but she must use at least one Rs.20 note. So, this is not possible.

So, there are 3 ways X can give Rs.500 to Y using only Rs.100 and Rs.20 notes, with the condition that she has only 16 notes of Rs.20, and being asked to use the notes of both the denominations.