3 integers are chosen at random from the set of first 20 natural numbers. The chance that their product is a multiple of 3, is.

Options

A. `194/285`
B. `1/57`
C. `13/19`
D. `3/4`

Answer

To solve the problem, we need to find the probability that the product of 3 randomly chosen integers from the first 20 natural numbers is a multiple of 3.

### Step-by-step Solution:

1. **Identify the Total Outcomes**:
We need to calculate the total number of ways to choose 3 integers from the first 20 natural numbers. This can be calculated using the combination formula:
\[ \text{Total ways} = \binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140 \]

**Hint**: Use the combination formula \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) to find the total ways to choose \(r\) items from \(n\).

2. **Identify the Favorable Outcomes**:
We need to find the number of ways to choose 3 integers such that their product is a multiple of 3. This occurs if at least one of the chosen integers is a multiple of 3.

3. **Count the Multiples of 3**:
The multiples of 3 from 1 to 20 are: 3, 6, 9, 12, 15, 18. There are a total of 6 multiples of 3.

4. **Count the Non-Multiples of 3**:
The total numbers from 1 to 20 is 20. Therefore, the non-multiples of 3 are:
\[ 20 - 6 = 14 \]

5. **Calculate the Cases**:
We will use complementary counting to find the number of ways to choose 3 integers where none of them is a multiple of 3.

- **Case 1**: Choose 3 integers from the 14 non-multiples of 3:
\[ \text{Ways} = \binom{14}{3} = \frac{14 \times 13 \times 12}{3 \times 2 \times 1} = 364 \]

6. **Calculate the Favorable Outcomes**:
The number of ways to choose 3 integers such that at least one is a multiple of 3 is:
\[ \text{Favorable outcomes} = \text{Total outcomes} - \text{Non-multiples outcomes} = 1140 - 364 = 776 \]

7. **Calculate the Probability**:
The probability that the product of the chosen integers is a multiple of 3 is given by:
\[ P(\text{multiple of 3}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{776}{1140} \]

8. **Simplify the Probability**:
To simplify \(\frac{776}{1140}\):
- Find the GCD of 776 and 1140, which is 4.
- Dividing both the numerator and denominator by 4 gives:
\[ \frac{776 \div 4}{1140 \div 4} = \frac{194}{285} \]

### Final Answer:
The probability that the product of the 3 chosen integers is a multiple of 3 is:
\[ \frac{194}{285} \]