A child has three different kinds of chocolates costing Rs. 2, Rs. 5 and Rs. 10. He spends total Rs. 120 on the chocolates. What is the minimum possible number of chocolates, he can buy, if there must be atleast one chocolate of each kind?

Options

A. 22
B. 19
C. 17
D. 15

Answer

To solve the problem, we need to determine the minimum number of chocolates the child can buy while ensuring that he buys at least one of each kind, given their respective costs of Rs. 2, Rs. 5, and Rs. 10, and that he spends a total of Rs. 120.

### Step-by-Step Solution:

1. **Identify the minimum requirement**: The child must buy at least one chocolate of each kind. Therefore, he will buy:
- 1 chocolate costing Rs. 2
- 1 chocolate costing Rs. 5
- 1 chocolate costing Rs. 10

**Total cost for the minimum chocolates**:
\[ 2 + 5 + 10 = 17 \text{ Rs.} \]

2. **Calculate remaining money**: After buying the minimum required chocolates, we subtract the total cost from the total money he has:
\[ 120 - 17 = 103 \text{ Rs.} \]

3. **Maximize the number of chocolates**: To minimize the total number of chocolates, we should buy as many of the cheapest chocolate (Rs. 2) as possible with the remaining money.

4. **Calculate how many Rs. 2 chocolates can be bought**:
\[ \text{Number of Rs. 2 chocolates} = \frac{103}{2} = 51.5 \]
Since he can only buy whole chocolates, he can buy 51 chocolates.

5. **Total chocolates bought**: Now, we add the chocolates bought:
- 51 chocolates of Rs. 2
- 1 chocolate of Rs. 5
- 1 chocolate of Rs. 10

**Total number of chocolates**:
\[ 51 + 1 + 1 = 53 \]

6. **Final check on total cost**: Let's check if the total cost adds up to Rs. 120:
- Cost of Rs. 2 chocolates: \(51 \times 2 = 102\)
- Cost of Rs. 5 chocolate: \(1 \times 5 = 5\)
- Cost of Rs. 10 chocolate: \(1 \times 10 = 10\)

**Total cost**:
\[ 102 + 5 + 10 = 117 \text{ Rs.} \]
Since we have Rs. 3 left, we can buy one more chocolate of Rs. 2.

7. **Final count of chocolates**:
- Now, he buys 52 chocolates of Rs. 2, 1 chocolate of Rs. 5, and 1 chocolate of Rs. 10.

**Final total**:
\[ 52 + 1 + 1 = 54 \]

### Conclusion:
The minimum possible number of chocolates he can buy is **54**.