Three numbers are chosen at random from 1 to 20 .The probability that they are consecutive is

Options

A. `1/190`
B. `1/120`
C. `3/190`
D. `5/190`

Answer

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

### Step 1: Calculate the Total Number of Ways to Choose 3 Numbers
The total number of ways to choose 3 numbers from a set of 20 can be calculated using the combination formula:

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

Calculating this gives:

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

### Step 2: Calculate the Number of Favorable Outcomes
Next, we need to find the number of ways to choose 3 consecutive numbers from the set {1, 2, ..., 20}.

The possible sets of three consecutive numbers are:
- (1, 2, 3)
- (2, 3, 4)
- (3, 4, 5)
- ...
- (18, 19, 20)

The last set of three consecutive numbers is (18, 19, 20).

To find the total number of such sets, we can see that the first number of the consecutive triplet can range from 1 to 18. Thus, there are 18 possible sets of three consecutive numbers.

### Step 3: Calculate the Probability
The probability of choosing 3 consecutive numbers is given by the ratio of the number of favorable outcomes to the total number of outcomes:

\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{18}{1140} \]

### Step 4: Simplify the Probability
Now, we can simplify the fraction:

\[ \frac{18}{1140} = \frac{3}{190} \]

Thus, the probability that three numbers chosen at random from 1 to 20 are consecutive is:

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

### Final Answer
The final answer is:

\[ \text{Probability} = \frac{3}{190} \]

---