Consider a hypothetical example.

There are [math]12[/math] tickets for a documentary film and exactly [math]12[/math] people to buy the tickets (one person can buy only one ticket).

There are [math]3[/math] ticketbooths for selling the tickets and each ticketbooth will sell exactly [math]4[/math] tickets.

Suppose the researcher labels the [math]12[/math] people with ID [math]1,2,\ldots, 12[/math] according to their arrivals.

Say, the first arrival buys his/her ticket from one of the [math]3[/math] ticketbooths.

Then, the second arrival buys his/her ticket from one of the [math]3[/math] ticketbooths. The second arrival can buy his/her ticket from the same ticket booth that the first arrival had bought or from one of the other two ticketbooths.

In this way, the last arrival buys his/her ticket.

The researcher has recorded from which ticketbooth which arrivals have bought the tickets.

Suppose from ticketbooth A, arrival #3, #5, #6, #12 have bought the tickets.

From ticketbooth B, arrival #1, #2, #9, #11 have bought the tickets.

From ticketbooth C, arrival #4, #7, #8, #10 have bought the tickets.

In how many ways the [math]12[/math] people can buy tickets from the [math]3[/math] ticketbooths?

#My Attempt:

If inside a ticketbooth the order of the ID doen't matter, then the number of ways the [math]12[/math] people can buy tickets from the [math]3[/math] ticketbooths is [math]=\frac{12!}{4!4!4!}.[/math]

But for my example, since inside a ticketbooth it is naturally ordered (that is, the first arrival comes before the second arrival and so on), should I consider the order inside a ticketbooth?

For the above example, is ORDER more meaningful or NOT?

There are [math]12[/math] tickets for a documentary film and exactly [math]12[/math] people to buy the tickets (one person can buy only one ticket).

There are [math]3[/math] ticketbooths for selling the tickets and each ticketbooth will sell exactly [math]4[/math] tickets.

Suppose the researcher labels the [math]12[/math] people with ID [math]1,2,\ldots, 12[/math] according to their arrivals.

Say, the first arrival buys his/her ticket from one of the [math]3[/math] ticketbooths.

Then, the second arrival buys his/her ticket from one of the [math]3[/math] ticketbooths. The second arrival can buy his/her ticket from the same ticket booth that the first arrival had bought or from one of the other two ticketbooths.

In this way, the last arrival buys his/her ticket.

The researcher has recorded from which ticketbooth which arrivals have bought the tickets.

Suppose from ticketbooth A, arrival #3, #5, #6, #12 have bought the tickets.

From ticketbooth B, arrival #1, #2, #9, #11 have bought the tickets.

From ticketbooth C, arrival #4, #7, #8, #10 have bought the tickets.

In how many ways the [math]12[/math] people can buy tickets from the [math]3[/math] ticketbooths?

#My Attempt:

If inside a ticketbooth the order of the ID doen't matter, then the number of ways the [math]12[/math] people can buy tickets from the [math]3[/math] ticketbooths is [math]=\frac{12!}{4!4!4!}.[/math]

But for my example, since inside a ticketbooth it is naturally ordered (that is, the first arrival comes before the second arrival and so on), should I consider the order inside a ticketbooth?

For the above example, is ORDER more meaningful or NOT?

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