Why is this $6!$ factorial and not $p(6,1)$?

Question

There is this question. There are six different candidates for governor of a state. In how many different orders can the names if the candidates be printed on a ballot?

The answer is $6!=720$. But why?

If there are $6$ candidates and order matters, they are being placed on $1$ card. Wouldn't it be $p(6,1)=6$ ways?

Answer 1

The given answer is correct.

You have $6$ choices for the first name.

For each of the $6$ choices, you have $5$ choices for the second name, and so on.

So, by the multiplication principle, we have that there are $6!$ ways to print the names.

Answer 2

$P(6,1) = 6$ is the number of ways you can select a candidate appointed as a governor while

$6! = P(6,6) = 720$ is the number of ways you arrange the $6$ names on a given ballot.

Answer 3

By the reasoning suggested in the question [which is incorrect, I must add, in order not to be misunderstood in the way in which I have already be misunderstood in a comment below.], if there were four candidates, named $A,B,C,D$, then there would be four orders in which they can be listed. Let's see if we can find all four: \begin{align} ABCD \\ ABDC \\ ACBD \\ ACDB \\ ADBC \\ ADCB \\[8pt] BACD \\ BADC \\ BCAD \\ BCDA \\ BDAC \\ BDCA \\[8pt] CABD \\ CADB \\ CBAD \\ CBDA \\ CDAB \\ CDBA \\[8pt] DABC \\ DACB \\ DBAC \\ DBCA \\ DCAB \\ DCBA \end{align} Now count them and see whether the answer is closer to $4$ or to $4!=24$.