I have a math question that's been boggling my mind, and I'm hoping someone can help me out. Here's the scenario:
The candidates A and B are the finalist for an award. A committee of 14 members will place their vote (in favour of one of the candidates) into a ballot box. Suppose that A receives 9 votes and B receives 5. In how many ways the ballot can be selected, one at a time, from the ballot box so that there are always more votes in favour of A. Explain your solution and print all such ways in Mathematica.
I attempted to solve this problem using Bertrand's Ballot Theorem, which states that the probability of A being strictly ahead of B throughout the count is given by $\frac{p-q}{p+q}$, where $p$ is the number of votes for A and $q$ is the number of votes for B. In our case, this would be $\frac{9-5}{9+5} = \frac{4}{14}$. But what next?
However, I'm struggling to figure out how many ways we can select a ballot, one at a time, from the ballot box, while maintaining the condition that A always has more votes than B in each selection. I'm not sure if the probability I calculated using Bertrand's theorem applies directly to counting the ways to select the ballots.
I would greatly appreciate it if someone could help me solve this problem and, if possible, provide all the ways these ballots can be selected while meeting the condition mentioned above. Please explain your solution clearly so that I can understand the process.