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)`

Explanation

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: Determine the total number of ways to choose 3 digits
We have 9 digits (1 to 9) and we need to choose 3 without repetition. The number of ways to choose 3 digits from 9 is given by the permutation formula:

\[ P(n, r) = \frac{n!}{(n-r)!} \]

Where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

For our case:

\[ P(9, 3) = \frac{9!}{(9-3)!} = \frac{9!}{6!} = 9 \times 8 \times 7 = 504 \]

### Step 2: Identify the conditions for the product to be odd
The product of three digits will be odd only if all three digits chosen are odd. The odd digits in our set are {1, 3, 5, 7, 9}.

### Step 3: Count the number of odd digits
There are 5 odd digits: 1, 3, 5, 7, and 9.

### Step 4: Calculate the number of ways to choose 3 odd digits
Using the same permutation formula, we can find the number of ways to choose 3 odd digits from the 5 available:

\[ P(5, 3) = \frac{5!}{(5-3)!} = \frac{5!}{2!} = 5 \times 4 \times 3 = 60 \]

### Step 5: Calculate the probability
The probability that the product of the chosen digits is odd is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P(\text{product is odd}) = \frac{\text{Number of ways to choose 3 odd digits}}{\text{Total number of ways to choose 3 digits}} = \frac{60}{504} \]

### Step 6: Simplify the probability
Now, we can simplify the fraction:

\[ \frac{60}{504} = \frac{5}{42} \]

### Final Answer
Thus, the probability that the product of the three chosen digits is odd is:

\[ \frac{5}{42} \]