Exploring the Permutations and Combinations for Parking Cars in Vacant Slots

Imagine a scenario where you have 8 vacant parking slots arranged in a row. You want to park exactly 6 cars in these slots. The question is, in how many ways can you do this? This problem can be broken down into two main steps: choosing the slots and arranging the cars within those slots.

Choosing the Slots

To start, we need to determine how many ways we can choose 6 out of the 8 slots for our cars. This is a combination problem, as we are selecting a subset of slots without regard to their order. The formula for combinations is given by:

[binom{n}{r} frac{n!}{r!(n-r)!}]

In this case, (n 8) (total slots) and (r 6) (slots to choose). Therefore, we calculate:

[binom{8}{6} frac{8!}{6!2!} frac{8 times 7}{2 times 1} 28]

Arranging the Cars in Chosen Slots

Once we have chosen the 6 slots, we need to consider the number of ways to arrange the 6 cars within these slots. Since the order in which the cars are placed matters, this is a permutation problem. The number of ways to arrange 6 cars is given by the factorial of 6, which is:

[6! 720]

Total Arrangements

To find the total number of ways to park the cars, we multiply the number of ways to choose the slots by the number of arrangements of the cars in those slots:

[text{Total Ways} binom{8}{6} times 6! 28 times 720 20160]

Therefore, the total number of ways to park 6 cars in 8 vacant parking slots is 20,160.

Contextual Considerations

A commenter mentioned that the cars can only occupy one of three positions, suggesting a constraint not explicitly stated in the original problem. If the problem were indeed constrained to specific positions, such as in a 2x4 grid, the solution would change. However, the problem as presented does not contain such constraints, and the solution remains 20,160.

Conclusion

The exploration of parking cars in vacant slots demonstrates the application of combinatorial mathematics. By understanding both combinations and permutations, we can solve complex problems involving arrangements of objects. In this case, understanding that cars can be arranged in different slots and that the order matters allows us to calculate the total number of possible arrangements.