Tired of being CIS 1600 TAs, Kevin, Daniel, Sophia, Suzzy, and Olivia train for the new 5x1600m
race in the 2028 Summer Olympics. However, they each want to be the first person to run, so
they establish a procedure to determine the order of the competitors. Their coach, Dilini, has
them line up in alphabetical order and tells them the following: the first two people must flip a
coin to decide who gets to be the first person in the relay. The winner of the coin flip is assigned
to the first spot in the relay, while the loser must stay to play the next person in line. The
person who wins the next flip gets to run second, while the loser must again stay to play the
next person in line. This process continues until there is only 1 TA left, who is the anchor. Basically, the winner stays, the loser continues. How many ways can the five TAs run?

Let's analyze the problem by breaking it down step by step.

The TAs are arranged alphabetically as follows: **Daniel, Kevin, Olivia, Sophia, Suzzy**. The process for determining the order of the runners is as follows:

1. **Daniel and Kevin** flip a coin.
   - The winner goes to the first spot.
   - The loser continues to the next round against Olivia.

2. The loser of the previous round faces **Olivia**.
   - The winner of this round goes to the second spot.
   - The loser continues to the next round against Sophia.

3. The loser of the previous round faces **Sophia**.
   - The winner of this round goes to the third spot.
   - The loser continues to the next round against Suzzy.

4. The loser of the previous round faces **Suzzy**.
   - The winner of this round goes to the fourth spot.
   - The loser becomes the anchor, filling the fifth spot.

### Counting the Number of Possible Orders

Let's consider the problem for \( n = 5 \) TAs. We need to determine the number of distinct possible orders of the runners that this procedure can generate.

At each stage, the loser continues to the next round. However, after each round, one runner is eliminated and placed in the order. We can model the situation as a series of choices, with each choice being the outcome of a coin flip.

For \( n \) people, the number of possible orders is given by \( 2^{n-1} \). This is because the process can be viewed as a binary tree where each node represents a coin flip, and each leaf node represents a possible ordering of runners.

#### Calculation for 5 TAs:
- For 5 people, the number of possible sequences is \( 2^{5-1} = 2^4 = 16 \).

Thus, there are **16 distinct ways** the five TAs can run the relay according to the rules described.

Would you like to go over more details or have any other questions?

### Relative Questions:
1. How does the coin flip affect the possible permutations if the number of participants changes?
2. What would happen if a tie-breaking rule was introduced for the final round?
3. How would the probability of each specific order be calculated?
4. How does this problem relate to other combinatorial selection processes?
5. Could the sequence be altered if the order of initial pairing changed?

### Tip:
Remember, when solving problems involving selections or arrangements, always consider if there's an underlying pattern or formula that can simplify the calculations.

Explore how to determine the order of 5 TAs in a relay race using coin flips. Learn about combinatorics, probability, and permutations with detailed step-by-step analysis. Discover the number of possible sequences and the mathematical principles involved.

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