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 form a triangle?

Options

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

Answer

To solve the problem of finding the probability that three randomly selected sticks can form a triangle, we will follow these steps:

### Step 1: Understand the Triangle Inequality Theorem
To determine if three lengths can form a triangle, we need to apply the Triangle Inequality Theorem. This theorem states that for any three sides \(a\), \(b\), and \(c\), the following conditions must be satisfied:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

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

### Step 3: Determine All Possible Combinations of 3 Sticks
We need to find all possible combinations of 3 sticks from the 5 given sticks. The total number of ways to choose 3 sticks from 5 can be calculated using the combination formula:
\[ \text{Number of combinations} = \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \(n\) is the total number of sticks and \(r\) is the number of sticks to choose.

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

### Step 4: Identify Valid Combinations That Can Form a Triangle
Now we will check each combination of 3 sticks to see if they satisfy the triangle inequality conditions.

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

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

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

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

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

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

7. **Combination: (3, 5, 7)**
- \(3 + 5 = 8 > 7\)
- \(3 + 7 = 10 > 5\)
- \(5 + 7 = 12 > 3\)
- **Can form a triangle.**

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

9. **Combination: (3, 7, 9)**
- \(3 + 7 = 10 > 9\)
- \(3 + 9 = 12 > 7\)
- \(7 + 9 = 16 > 3\)
- **Can form a triangle.**

10. **Combination: (5, 7, 9)**
- \(5 + 7 = 12 > 9\)
- \(5 + 9 = 14 > 7\)
- \(7 + 9 = 16 > 5\)
- **Can form a triangle.**

### Step 5: Count the Valid Combinations
From the combinations checked, the valid combinations that can form a triangle are:
- (3, 5, 7)
- (3, 7, 9)
- (5, 7, 9)

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

### Step 6: Calculate the Probability
The probability \(P\) that the selected sticks can form a triangle is given by:
\[ P = \frac{\text{Number of valid combinations}}{\text{Total combinations}} = \frac{3}{10} = 0.3 \]

### Final Answer
The probability that the selected sticks can form a triangle is **0.3**.

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