A town is voting on an ordinance dealing with smoking in public spaces. The options are (A) permit unrestricted smoking; $(B)$ permit smoking in designated areas only; and $(\mathrm{C})$ ban all smoking in public places. The winner is to be determined by the Borda count method. The preference table for the election is shown. \begin{tabular}{|c|c|c|c|c|c|} \hline $\begin{array}{c}\text { Number of } \\ \text { Votes }\end{array}$ & 156 & 78 & 39 & 39 & 39 \\ \hline 1st Choice & A & C & B & C & B \\ \hline $\begin{array}{c}\text { 2nd } \\ \text { Choice }\end{array}$ & C & B & A & A & C \\ \hline 3rd Choice & B & A & C & B & A \\ \hline \end{tabular} A. Which option is favored over all others using a head-to-head comparison? B. Which option wins the vote using the Borda count method? C. Is the head-to-head criterion satisfied? Why or why not? A. The option favored over all others using a head-to-head comparison is option B. The option that wins the vote using the Borda count method is option C. Is the head-to-head criterion satisfied? Why or why not?

Step 1 ：First, we need to understand the Borda count method. In this method, each voter ranks the options in order of preference. The first choice of each voter gets a number of points equal to the number of options, the second choice gets one point less, and so on, until the last choice gets one point. The option with the most points wins.

Step 2 ：To answer the first question, we need to compare each option against each other option in a head-to-head comparison. This means we need to count how many voters prefer each option to each other option.

Step 3 ：To answer the second question, we need to calculate the Borda count for each option. This means we need to multiply the number of votes each option gets in each position by the number of points that position is worth, and then sum these products for each option.

Step 4 ：To answer the third question, we need to check if the option that wins the head-to-head comparison also wins the Borda count. If it does, the head-to-head criterion is satisfied.

Step 5 ：The Borda count for each option is calculated as follows: 'A': 741, 'B': 585, 'C': 780

Step 6 ：The head-to-head count for each option is calculated as follows: 'A': 390, 'B': 234, 'C': 429

Step 7 ：The option favored over all others using a head-to-head comparison is option \(\boxed{C}\).

Step 8 ：The option that wins the vote using the Borda count method is option \(\boxed{C}\).

Step 9 ：The head-to-head criterion is satisfied because the option that wins the head-to-head comparison also wins the Borda count.