The number of ways in which we can select four numbers from 1 to 30 so as to exclude every selection of four consecutive words is

Options

A. `"^30 C_4-20`
B. `"^30 C_4-22`
C. `"^30C_4-27`
D. none of these

Answer

To solve the problem of selecting four numbers from 1 to 30 while excluding any selection of four consecutive numbers, we can use a combinatorial approach. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the Problem
We need to select 4 numbers from the set {1, 2, ..., 30} such that no four selected numbers are consecutive.

### Step 2: Adjust the Selection
To ensure that no four selected numbers are consecutive, we can transform the problem. If we select a number, we need to "block" the next three numbers from being selected.

### Step 3: Create New Variables
Let’s define new variables to account for the blocked numbers. If we select a number \( x_1 \), then the next possible number we can select would be \( x_1 + 4 \) (i.e., skipping the next three numbers).

### Step 4: Define the New Selection
Let’s define new variables:
- Let \( y_1 = x_1 \)
- Let \( y_2 = x_2 - 1 \)
- Let \( y_3 = x_3 - 2 \)
- Let \( y_4 = x_4 - 3 \)

This transformation shifts the selection to ensure that the numbers are not consecutive.

### Step 5: Set Up the Equation
Now, we need to select 4 numbers \( y_1, y_2, y_3, y_4 \) from the set {1, 2, ..., 30 - 3} = {1, 2, ..., 27}. This is because we have effectively reduced the range by 3 due to the shifts.

### Step 6: Calculate the Combinations
Now, we need to find the number of ways to choose 4 numbers from 27. This can be calculated using the combination formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n = 27 \) and \( r = 4 \).

### Step 7: Compute the Value
Calculating \( \binom{27}{4} \):
\[ \binom{27}{4} = \frac{27!}{4!(27-4)!} = \frac{27 \times 26 \times 25 \times 24}{4 \times 3 \times 2 \times 1} \]
Calculating the numerator:
\[ 27 \times 26 = 702, \quad 702 \times 25 = 17550, \quad 17550 \times 24 = 421200 \]
Calculating the denominator:
\[ 4 \times 3 \times 2 \times 1 = 24 \]
Now divide:
\[ \frac{421200}{24} = 17550 \]

### Final Answer
Thus, the number of ways to select 4 numbers from 1 to 30 while excluding any selection of four consecutive numbers is **17550**.

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