Three numbers are chosen from 1 to 20. Find the probability that they are consecutive.

Options

A. `(186)/(190)`
B. `(187)/(190)`
C. `(188)/(190)`
D. `(18)/(.^(20)C_(3))`

Explanation

To find the probability that three numbers chosen from 1 to 20 are consecutive, we can follow these steps:

### Step 1: Determine the total number of ways to choose 3 numbers from 20.
The total ways to choose 3 numbers from 20 can be calculated using the combination formula:

\[ \text{Total ways} = \binom{20}{3} = \frac{20!}{3!(20-3)!} = \frac{20!}{3! \cdot 17!} \]

### Step 2: Calculate the value of \(\binom{20}{3}\).
Calculating \(\binom{20}{3}\):

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

### Step 3: Determine the number of ways to choose 3 consecutive numbers.
The consecutive numbers can be chosen starting from 1 up to 18. The possible sets of consecutive numbers are:

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

Thus, the possible sets of consecutive numbers are:

\[ (1, 2, 3), (2, 3, 4), (3, 4, 5), \ldots, (18, 19, 20) \]

There are 18 sets of consecutive numbers.

### Step 4: Calculate the probability of choosing 3 consecutive numbers.
The probability \(P\) that the three chosen numbers are consecutive is given by the ratio of the number of favorable outcomes to the total outcomes:

\[ P = \frac{\text{Number of ways to choose 3 consecutive numbers}}{\text{Total ways to choose 3 numbers}} = \frac{18}{1140} \]

### Step 5: Simplify the probability.
Now, we simplify \(\frac{18}{1140}\):

\[ P = \frac{18 \div 6}{1140 \div 6} = \frac{3}{190} \]

### Final Answer:
Thus, the probability that three numbers chosen from 1 to 20 are consecutive is:

\[ \boxed{\frac{3}{190}} \]