Correct Option

There are 5 distinct denominations available: ₹50, ₹100, ₹200, ₹500, and ₹2000.

The problem requires forming different sums by selecting at least three denominations at a time. This implies considering combinations of 3, 4, or 5 denominations from the given 5.

The number of ways to choose 'r' items from 'n' distinct items is given by the combination formula:

$(^{n} C_{r} = \frac{n!}{r! (n - r)!})$

Number of ways to choose exactly 3 denominations: $5 C_{3} = \frac{5!}{3! (5 - 3)!} = \frac{5!}{3! 2!} = \frac{5 \times 4}{2 \times 1} = 10$ .
Number of ways to choose exactly 4 denominations: $^{5} C_{4} = \frac{5!}{4! (5 - 4)!} = \frac{5!}{4! 1!} = 5$ .
Number of ways to choose exactly 5 denominations: $^{5} C_{5} = \frac{5!}{5! (5 - 5)!} = 1$ .

The total number of different sums is the sum of these possibilities:

Required different sums = $5 C_{3} +^{5} C_{4} +^{5} C_{5}$ = 10 + 5 + 1 = 16.

Incorrect Options

Option 2 (15): This value would be obtained if only combinations of 3 and 4 denominations were considered ( $5 C_{3} +^{5} C_{4} = 10 + 5 = 15$ ), omitting the case where all 5 denominations are selected.
Option 3 (14): This value does not correspond to a direct calculation based on the problem's criteria for selecting at least three denominations.
Option 4 (10): This value represents only the number of ways to choose exactly 3 denominations ( $5 C_{3} = 10$ ), failing to account for combinations of 4 or 5 denominations as required by the "at least three" condition.