Three numbers are chosen from 1 to 30. The probability that they are not consecutive is

Options

A. `(144)/(145)`
B. `(143)/(145)`
C. `(142)/(145)`
D. none of these

Answer

To find the probability that three numbers chosen from 1 to 30 are not consecutive, we can follow these steps:

### Step 1: Calculate the total number of ways to choose 3 numbers from 30.
The total number of ways to choose 3 numbers from 30 is given by the combination formula:

\[ \text{Total ways} = \binom{30}{3} = \frac{30!}{3!(30-3)!} = \frac{30 \times 29 \times 28}{3 \times 2 \times 1} = 4060 \]

### Step 2: Calculate the number of ways to choose 3 consecutive numbers.
If we want to choose 3 consecutive numbers, we can think of them as a block. The smallest block starts from 1 (i.e., 1, 2, 3) and the largest block starts from 28 (i.e., 28, 29, 30).

The possible blocks of consecutive numbers are:
- (1, 2, 3)
- (2, 3, 4)
- (3, 4, 5)
- ...
- (28, 29, 30)

This gives us a total of 28 combinations of consecutive numbers.

### Step 3: Calculate the probability of choosing 3 consecutive numbers.
The probability \( P(A) \) that the three chosen numbers are consecutive is:

\[ P(A) = \frac{\text{Number of ways to choose 3 consecutive numbers}}{\text{Total ways to choose 3 numbers}} = \frac{28}{4060} \]

### Step 4: Simplify the probability of choosing consecutive numbers.
We can simplify \( \frac{28}{4060} \):

\[ P(A) = \frac{28}{4060} = \frac{1}{145} \]

### Step 5: Calculate the probability that the numbers are not consecutive.
The probability that the three chosen numbers are not consecutive is the complement of the probability that they are consecutive:

\[ P(A') = 1 - P(A) = 1 - \frac{1}{145} = \frac{144}{145} \]

### Final Answer:
Thus, the probability that the three numbers chosen from 1 to 30 are not consecutive is:

\[ \frac{144}{145} \]

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