Three dice are thrown. The probability of getting a sum which is a perfect square, is

Options

A. `2//5`
B. `9//20`
C. `1//4`
D. none of these

Explanation

To find the probability of getting a sum that is a perfect square when three dice are thrown, we can follow these steps:

### Step 1: Identify the possible sums
When three dice are thrown, the minimum sum is 3 (1+1+1) and the maximum sum is 18 (6+6+6). The possible sums that are perfect squares within this range are 4, 9, and 16.

### Step 2: Calculate the total number of outcomes
The total number of outcomes when throwing three dice is given by:
\[ 6 \times 6 \times 6 = 216 \]

### Step 3: Calculate the number of favorable outcomes for each perfect square sum

#### For a sum of 4:
The combinations that yield a sum of 4 are:
- (1, 1, 2)
- (1, 2, 1)
- (2, 1, 1)

Thus, there are 3 combinations.

#### For a sum of 9:
The combinations that yield a sum of 9 can be calculated systematically or by enumeration. The valid combinations are:
- (1, 2, 6)
- (1, 3, 5)
- (1, 4, 4)
- (2, 1, 6)
- (2, 2, 5)
- (2, 3, 4)
- (3, 1, 5)
- (3, 2, 4)
- (3, 3, 3)
- (4, 1, 4)
- (4, 2, 3)
- (5, 1, 3)
- (5, 2, 2)
- (6, 1, 2)

After counting all combinations, we find there are 25 combinations that yield a sum of 9.

#### For a sum of 16:
The combinations that yield a sum of 16 are:
- (6, 6, 4)
- (6, 5, 5)
- (5, 6, 5)
- (4, 6, 6)

Thus, there are 7 combinations.

### Step 4: Calculate the total number of favorable outcomes
Now we can sum the number of favorable outcomes:
- For a sum of 4: 3 outcomes
- For a sum of 9: 25 outcomes
- For a sum of 16: 7 outcomes

Total favorable outcomes = 3 + 25 + 7 = 35

### Step 5: Calculate the probability
The probability \( P \) of getting a sum that is a perfect square is given by the ratio of favorable outcomes to total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{35}{216} \]

### Final Answer
The probability of getting a sum which is a perfect square when three dice are thrown is:
\[ \frac{35}{216} \]