Five sticks of length 1, 3, 5, 7 and 9 feet are given. Three of these sticks are selected at random. What is the probability that the selected sticks can from a triangle?

Options

A. 0.5
B. 0.4
C. 0.3
D. 0

Explanation

To solve the problem of finding the probability that three randomly selected sticks from lengths 1, 3, 5, 7, and 9 feet can form a triangle, we will follow these steps:

### Step 1: Understand the Triangle Inequality Theorem
To form a triangle with three sides of lengths \(a\), \(b\), and \(c\), the following conditions must hold:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

For our purpose, we can simplify this by ensuring that the sum of the two shorter sides is greater than the longest side.

### Step 2: List the Sticks
The lengths of the sticks are:
- Stick 1: 1 foot
- Stick 2: 3 feet
- Stick 3: 5 feet
- Stick 4: 7 feet
- Stick 5: 9 feet

### Step 3: Calculate the Total Combinations
We need to find the total number of ways to choose 3 sticks from the 5 available. This can be calculated using the combination formula:
\[ \text{Total combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
Where \(n\) is the total number of items (sticks) and \(r\) is the number of items to choose.

For our case:
\[ \text{Total combinations} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Identify Favorable Combinations
Now, we will check which combinations of three sticks satisfy the triangle inequality.

1. **Combination (1, 3, 5)**:
- \(1 + 3 = 4\) (not greater than 5)
- **Does not form a triangle**.

2. **Combination (1, 3, 7)**:
- \(1 + 3 = 4\) (not greater than 7)
- **Does not form a triangle**.

3. **Combination (1, 3, 9)**:
- \(1 + 3 = 4\) (not greater than 9)
- **Does not form a triangle**.

4. **Combination (1, 5, 7)**:
- \(1 + 5 = 6\) (not greater than 7)
- **Does not form a triangle**.

5. **Combination (1, 5, 9)**:
- \(1 + 5 = 6\) (not greater than 9)
- **Does not form a triangle**.

6. **Combination (1, 7, 9)**:
- \(1 + 7 = 8\) (not greater than 9)
- **Does not form a triangle**.

7. **Combination (3, 5, 7)**:
- \(3 + 5 = 8\) (greater than 7)
- **Forms a triangle**.

8. **Combination (3, 5, 9)**:
- \(3 + 5 = 8\) (not greater than 9)
- **Does not form a triangle**.

9. **Combination (3, 7, 9)**:
- \(3 + 7 = 10\) (greater than 9)
- **Forms a triangle**.

10. **Combination (5, 7, 9)**:
- \(5 + 7 = 12\) (greater than 9)
- **Forms a triangle**.

### Step 5: Count Favorable Outcomes
From the combinations checked, the favorable combinations that can form triangles are:
- (3, 5, 7)
- (3, 7, 9)
- (5, 7, 9)

Thus, there are **3 favorable combinations**.

### Step 6: Calculate the Probability
The probability \(P\) that the selected sticks can form a triangle is given by the ratio of the number of favorable outcomes to the total outcomes:
\[ P = \frac{\text{Number of favorable outcomes}}{\text{Total combinations}} = \frac{3}{10} \]

### Final Answer
Thus, the probability that the selected sticks can form a triangle is:
\[ \boxed{0.3} \]