Number of Ways to Schedule Non Consecutive Factory Visits

To find the total number of possible ways to schedule the factory visits, we can formulate this as a problem of selecting non-consecutive days from a set of $15$ days.

Step 1: Identify the components

• Total number of days available, $n = 15$.
• Number of visit days, $k = 4$.
• Number of non-visit days = $15 - 4 = 11$.

Step 2: Apply the gap method Let $V$ represent a day with a factory visit and $N$ represent a day without a factory visit.

First, arrange the $11$ non-visit days ($N$) in a line: $\text{\_ } N_1 \text{ \_ } N_2 \text{ \_ } N_3 \text{ \_ } \dots \text{ \_ } N_{11} \text{ \_}$

These $11$ non-visit days create a total of $11 + 1 = 12$ distinct gaps (including the spaces before the first non-visit day and after the last non-visit day).

Step 3: Choose positions for the visit days To ensure that no two visit days ($V$) occur on consecutive days, we must place at most one visit day in each available gap. Therefore, we need to choose $4$ gaps out of the $12$ available gaps to place the $4$ visit days.

The number of ways to select $4$ gaps out of $12$ is given by the combination formula: $\text{Number of ways} = \binom{12}{4}$

Step 4: Calculation $\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1}$

$\binom{12}{4} = \frac{12 \times 11 \times 10 \times 9}{24} = 11 \times 5 \times 9 = 495$

Thus, the total number of possible ways to schedule these visits is 495.