Three numbers are chosen at random from the first 20 natural numbers. The probability that they are not consecutive is

Options

A. `187/190`
B. `93/95`
C. `94/95`
D. `3/190`

Answer

To find the probability that three numbers chosen at random from the first 20 natural numbers are not consecutive, we can follow these steps:

### Step 1: Total Ways to Choose 3 Numbers
First, we need to calculate the total number of ways to choose 3 numbers from the first 20 natural numbers. This can be done using the combination formula:

\[ \text{Total ways} = \binom{20}{3} = \frac{20 \times 19 \times 18}{3 \times 2 \times 1} = 1140 \]

### Step 2: Ways to Choose 3 Consecutive Numbers
Next, we need to find the number of ways to choose 3 consecutive numbers from the first 20 natural numbers. The sets of 3 consecutive numbers are:

- (1, 2, 3)
- (2, 3, 4)
- (3, 4, 5)
- ...
- (18, 19, 20)

The last set is (18, 19, 20), and we can see that there are 18 such sets. Therefore, the number of ways to choose 3 consecutive numbers is 18.

### Step 3: Probability of Choosing Consecutive Numbers
Now, we can find the probability that the chosen numbers are consecutive:

\[ P(\text{consecutive}) = \frac{\text{Number of ways to choose consecutive numbers}}{\text{Total ways}} = \frac{18}{1140} \]

We can simplify this fraction:

\[ P(\text{consecutive}) = \frac{3}{190} \]

### Step 4: Probability of Choosing Non-Consecutive Numbers
To find the probability that the chosen numbers are not consecutive, we subtract the probability of choosing consecutive numbers from 1:

\[ P(\text{not consecutive}) = 1 - P(\text{consecutive}) = 1 - \frac{3}{190} \]

Calculating this gives:

\[ P(\text{not consecutive}) = \frac{190 - 3}{190} = \frac{187}{190} \]

### Final Answer
Thus, the probability that the three numbers chosen are not consecutive is:

\[ \boxed{\frac{187}{190}} \]

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