Four numbers are chosen at random from `{1,2,3,......, 40}`. The probability that they are not consecutive is

Options

A. `(1)/(2470)`
B. `(4)/(7969)`
C. `(2469)/(2470)`
D. `(7965)/(7969)`

Answer

To solve the problem of finding the probability that four numbers chosen at random from the set {1, 2, 3, ..., 40} are not consecutive, we can follow these steps:

### Step 1: Calculate the Total Number of Outcomes
The total number of ways to choose 4 numbers from 40 is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to choose.

\[ \text{Total Outcomes} = \binom{40}{4} = \frac{40 \times 39 \times 38 \times 37}{4 \times 3 \times 2 \times 1} = 91,390 \]

### Step 2: Calculate the Number of Favorable Outcomes (Not Consecutive)
To find the number of ways to choose 4 numbers that are not consecutive, we can use the complementary counting method. First, we will calculate the number of ways to choose 4 consecutive numbers.

If we denote the 4 chosen numbers as \( x_1, x_2, x_3, x_4 \), and they are consecutive, we can represent them as:
- \( x_1 = a \)
- \( x_2 = a + 1 \)
- \( x_3 = a + 2 \)
- \( x_4 = a + 3 \)

The smallest value for \( a \) can be 1 (for the numbers 1, 2, 3, 4) and the largest value for \( a \) can be 37 (for the numbers 37, 38, 39, 40). Therefore, the possible values for \( a \) are from 1 to 37.

Thus, there are 37 ways to choose 4 consecutive numbers.

### Step 3: Calculate the Probability of Choosing Consecutive Numbers
Now, we can find the probability of choosing 4 consecutive numbers:

\[ P(\text{Consecutive}) = \frac{\text{Number of ways to choose consecutive numbers}}{\text{Total outcomes}} = \frac{37}{91,390} \]

### Step 4: Calculate the Probability of Choosing Non-Consecutive Numbers
To find the probability of choosing numbers that are not consecutive, we can use the complement:

\[ P(\text{Not Consecutive}) = 1 - P(\text{Consecutive}) = 1 - \frac{37}{91,390} \]

Calculating this gives:

\[ P(\text{Not Consecutive}) = 1 - \frac{37}{91,390} = \frac{91,390 - 37}{91,390} = \frac{91,353}{91,390} \]

### Step 5: Simplify the Probability
Now, we can simplify the fraction:

\[ P(\text{Not Consecutive}) = \frac{91,353}{91,390} \approx \frac{2469}{2470} \]

Thus, the final probability that the four chosen numbers are not consecutive is:

\[ \boxed{\frac{2469}{2470}} \]