How many different sums can be formed with the denominations ₹50, ₹100, ₹200, ₹500 and ₹2000 taking at least three denominations at a time?

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: $5C_3=\frac{5!}{3!(5-3)!}=\frac{5!}{3!2!}=\frac{5\times 4}{2\times 1}=10$.
• 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 = $5C_3{+}^5{C}_4{+}^5{C}_5$ = 10 + 5 + 1 = 16.

Incorrect Options

• Option 2 (15): This value would be obtained if only combinations of 3 and 4 denominations were considered ($5C_3{+}^5{C}_4=10+5=15$), 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 ($5C_3=10$), failing to account for combinations of 4 or 5 denominations as required by the "at least three" condition.