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

Options

A. `2/19`
B. `3/29`
C. `17/19`
D. `4/29`

Answer

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: Understand the Total Outcomes
We need to determine the total number of ways to choose 3 integers from the first 20 integers. This can be calculated using the combination formula:

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

### Step 2: Calculate Total Outcomes
Calculating the total number of combinations:

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

### Step 3: Determine the Favorable Outcomes for an Odd Product
Next, we need to find the number of ways to choose 3 integers such that their product is odd. For the product to be odd, all three chosen integers must be odd.

In the first 20 integers, there are 10 odd integers (1, 3, 5, 7, 9, 11, 13, 15, 17, 19).

We can calculate the number of ways to choose 3 odd integers:

\[ \text{Favorable Outcomes (Odd Product)} = \binom{10}{3} \]

### Step 4: Calculate Favorable Outcomes for Odd Product
Calculating the number of combinations of odd integers:

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

### Step 5: Calculate the Probability of an Odd Product
Now, we can calculate the probability that the product of the three integers is odd:

\[ P(\text{Odd Product}) = \frac{\text{Favorable Outcomes (Odd Product)}}{\text{Total Outcomes}} = \frac{120}{1140} \]

### Step 6: Simplify the Probability of an Odd Product
Simplifying the fraction:

\[ P(\text{Odd Product}) = \frac{120 \div 120}{1140 \div 120} = \frac{1}{9.5} = \frac{2}{19} \]

### Step 7: Calculate the Probability of an Even Product
The probability that the product is even is the complement of the probability that the product is odd:

\[ P(\text{Even Product}) = 1 - P(\text{Odd Product}) = 1 - \frac{2}{19} = \frac{19 - 2}{19} = \frac{17}{19} \]

### Final Answer
Thus, the probability that the product of the three integers is even is:

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