Arranging 6 People in a Line: Ensuring Tallest and Shortest Don't Stand Together

When arranging 6 people in a line, ensuring that the tallest and shortest individuals are not standing next to each other can be achieved through the principle of complementary counting. This method involves calculating the total number of arrangements and subtracting the number of arrangements where the tallest and shortest are together. Let's explore this step-by-step.

The Principle of Complementary Counting

Complementary counting is a useful technique in combinatorics where we find the number of desired outcomes by subtracting the number of undesired outcomes from the total number of outcomes. Here, we want to find the number of arrangements where the tallest and shortest are not together.

Total Arrangements Without Restrictions

The total number of ways to arrange 6 people in a line is given by the factorial of 6:

$$6! 720$$

Arrangements Where the Tallest and Shortest Are Together

To find the number of arrangements where the tallest and shortest, denoted as T and S, are together, we can treat them as a single unit or block. This reduces the problem to arranging 5 units (the combined T and S unit and the other 4 individuals).

The number of ways to arrange these 5 units is:

$$5! 120$$

Within the block T and S can be arranged in 2 ways (TS or ST). Therefore, the total number of arrangements where T and S are together is:

$$5! times 2 120 times 2 240$$

Subtracting Restricted Arrangements from Total Arrangements

Finally, to find the number of arrangements where the tallest and shortest are not standing next to each other, we subtract the number of restricted arrangements from the total number of arrangements:

$$720 - 240 480$$

Therefore, the number of ways to arrange 6 people in a line such that the tallest and shortest do not stand next to each other is 480.

Alternative Approach

The tallest and shortest can be placed in a certain number of ways. Specifically, they can be arranged in 5! / 3! ways by leaving the other 4 people to be placed in 4! ways. This can be represented as:

$$frac{5!}{3!4!} 480$$

Conclusion

By following the principle of complementary counting, we have determined that the number of ways to arrange 6 people in a line such that the tallest and shortest do not stand next to each other is 480. This method is particularly useful in solving similar combinatorial problems involving restrictions on the order of elements.