If the lengths of the sides of a triangle are decided by the three thrown of a single fair die,then the probability that the triangle is of maximum area given that it is an isosceles triangle, is

Options

A. `(1)/(7)`
B. `(1)/(27)`
C. `(1)/(14)`
D. None of these

Answer

To solve the problem, we need to find the probability that a triangle formed by the lengths obtained from three throws of a die is of maximum area, given that it is an isosceles triangle.

### Step-by-Step Solution:

1. **Identify the Sample Space**:
When rolling a die three times, the possible outcomes for the lengths of the sides of the triangle can be represented as ordered triples \((a, b, c)\), where \(a\), \(b\), and \(c\) are the results of the three rolls. The total number of outcomes when rolling a die three times is \(6^3 = 216\).

**Hint**: Consider how many different outcomes can be generated when rolling a die multiple times.

2. **Determine the Isosceles Triangle Condition**:
An isosceles triangle has at least two sides equal. We need to count the valid combinations of \((a, b, c)\) that satisfy this condition. The possible cases for isosceles triangles are:
- \(a = b \neq c\)
- \(a = c \neq b\)
- \(b = c \neq a\)

For each case, we can have the following combinations:
- For \(a = b\): \(1 \leq a \leq 6\) and \(c\) can be any value different from \(a\) (5 options).
- Therefore, there are \(6 \times 5 = 30\) combinations for \(a = b\).
- The same logic applies for the other two cases, leading to \(30\) combinations for each case.

Thus, the total number of isosceles triangles is \(30 + 30 + 30 = 90\).

**Hint**: Think about how to count the combinations where two sides are equal and the third is different.

3. **Identify Maximum Area Condition**:
For an isosceles triangle, the maximum area occurs when the triangle is equilateral. However, since we are limited to the outcomes of a die, the only equilateral triangle we can form is when all three sides are equal, i.e., \(a = b = c\). The possible outcomes for this scenario are:
- \( (1, 1, 1) \)
- \( (2, 2, 2) \)
- \( (3, 3, 3) \)
- \( (4, 4, 4) \)
- \( (5, 5, 5) \)
- \( (6, 6, 6) \)

This gives us \(6\) combinations of equilateral triangles.

**Hint**: Consider how many ways you can have all three sides equal when rolling a die.

4. **Calculate the Probability**:
The probability \(P\) that a randomly selected isosceles triangle is of maximum area (equilateral) is given by the ratio of the number of favorable outcomes (equilateral triangles) to the total number of isosceles triangles:

\[ P = \frac{\text{Number of equilateral triangles}}{\text{Total number of isosceles triangles}} = \frac{6}{90} = \frac{1}{15} \]

**Hint**: Remember to simplify the fraction to find the final probability.

### Final Answer:
The probability that the triangle is of maximum area given that it is an isosceles triangle is \(\frac{1}{15}\).