Three digits are chosen at random from 1, 2, 3, 4, 5, 6, 7, 8 and 9 without repeating any digit. What is the probability that the product is odd?

Options

A. `(2)/(3)`
B. `(7)/(48)`
C. `(5)/(42)`
D. `(5)/(108)`

Answer

To solve the problem of finding the probability that the product of three randomly chosen digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} is odd, we can follow these steps:

### Step 1: Identify the odd digits
The odd digits from the set {1, 2, 3, 4, 5, 6, 7, 8, 9} are:
- 1, 3, 5, 7, 9

There are a total of **5 odd digits**.

### Step 2: Determine the total number of ways to choose 3 digits
The total number of digits available is 9. We need to choose 3 digits from these 9 digits. The number of ways to choose 3 digits from 9 is given by the combination formula:
\[ \text{Total ways} = \binom{9}{3} = \frac{9!}{3!(9-3)!} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 84 \]

### Step 3: Determine the number of ways to choose 3 odd digits
To ensure the product of the digits is odd, all three chosen digits must be odd. The number of ways to choose 3 odd digits from the 5 available odd digits is:
\[ \text{Ways to choose odd digits} = \binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

### Step 4: Calculate the probability
The probability that the product of the chosen digits is odd is the ratio of the number of favorable outcomes (choosing 3 odd digits) to the total number of outcomes (choosing any 3 digits):
\[ \text{Probability} = \frac{\text{Ways to choose odd digits}}{\text{Total ways}} = \frac{10}{84} = \frac{5}{42} \]

### Final Answer
Thus, the probability that the product of the three chosen digits is odd is:
\[ \frac{5}{42} \]

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