Three integers are chosen at random from the first 20 integers. The probability that their product is even is:

Options

A. `4/19`
B. `17/19`
C. `16/19`
D. `5/19`

Explanation

To solve the problem of finding the probability that the product of three randomly chosen integers from the first 20 integers is even, we can follow these steps:

### Step 1: Determine the total number of ways to choose 3 integers from 20.
The total number of ways to choose 3 integers from 20 is given by the combination formula:

\[ \text{Total ways} = \binom{20}{3} \]

Calculating this:

\[ \binom{20}{3} = \frac{20!}{3!(20-3)!} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = \frac{6840}{6} = 1140 \]

### Step 2: Determine the number of odd integers in the first 20 integers.
In the first 20 integers, the odd integers are: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19. There are a total of 10 odd integers.

### Step 3: Calculate the number of ways to choose 3 odd integers.
The number of ways to choose 3 integers from these 10 odd integers is:

\[ \text{Ways to choose 3 odd integers} = \binom{10}{3} \]

Calculating this:

\[ \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = \frac{720}{6} = 120 \]

### Step 4: Calculate the probability that all chosen integers are odd.
The probability that all three chosen integers are odd is given by the ratio of the number of ways to choose 3 odd integers to the total number of ways to choose 3 integers:

\[ P(\text{all odd}) = \frac{\binom{10}{3}}{\binom{20}{3}} = \frac{120}{1140} \]

Simplifying this fraction:

\[ P(\text{all odd}) = \frac{1}{9.5} = \frac{2}{19} \]

### Step 5: Calculate the probability that at least one integer is even.
The probability that at least one of the integers is even is the complement of the probability that all are odd:

\[ P(\text{at least one even}) = 1 - P(\text{all odd}) = 1 - \frac{2}{19} \]

Calculating this:

\[ P(\text{at least one even}) = \frac{19}{19} - \frac{2}{19} = \frac{17}{19} \]

### Final Answer:
The probability that the product of the three chosen integers is even is:

\[ \frac{17}{19} \]