An academic department with five faculty members narrowed its choice for department head to either candidate A or candidate B. Each member then voted on a slip of paper for one of the candidates. Suppose there are actually three votes for A and two for B. If the slips are selected for tallying in random order, what is the probability that A remains ahead of B throughout the vote count (e.g., this event occurs if the selected ordering is AABAB, but not for ABBAA)?

We are tasked with finding the probability that candidate A remains ahead of candidate B throughout the vote count, given that A receives 3 votes and B receives 2 votes. 

This type of problem can be analyzed using concepts from combinatorics, specifically the **Catalan number** and the **Ballot Theorem**.

### Step-by-Step Solution:

#### Total number of possible vote sequences:
We are given that A gets 3 votes and B gets 2 votes. The total number of ways to arrange 3 A's and 2 B's is computed using the formula for combinations:
\[
\text{Total arrangements} = \binom{5}{3} = \frac{5!}{3!2!} = \frac{120}{6 \times 2} = 10.
\]
So, there are 10 different ways to arrange the votes.

#### Ballot Theorem:
According to the **Ballot Theorem**, the probability that, in an election with \(a\) votes for A and \(b\) votes for B, A remains ahead of B throughout the vote counting is given by:
\[
P(\text{A always ahead}) = \frac{a - b}{a + b}.
\]
In this case, \(a = 3\) and \(b = 2\), so the probability is:
\[
P(\text{A always ahead}) = \frac{3 - 2}{3 + 2} = \frac{1}{5}.
\]

Thus, the probability that A remains ahead of B throughout the vote count is **\(\frac{1}{5}\)**.

#### Conclusion:
The probability that A remains ahead of B throughout the vote counting process is \(\frac{1}{5}\).

Would you like more details or have any questions?

Here are 5 related questions for further exploration:
1. How would the probability change if A and B received 4 and 3 votes, respectively?
2. Can the Catalan number formula be applied to solve this problem directly?
3. How can the Ballot Theorem be proven formally using recursion or combinatorics?
4. What is the total number of vote sequences where A leads after the second vote count?
5. How does this problem relate to random walks and paths in graph theory?

**Tip**: When tackling combinatorial problems, breaking them down using known theorems like the Ballot Theorem can provide quick and accurate results.

Explore the probability that candidate A remains ahead of candidate B throughout the vote count using combinatorics and the Ballot Theorem. This problem involves 3 votes for A and 2 votes for B, with a detailed breakdown of the solution using key formulas. Ideal for students in Grades 11-12 and college-level courses.

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