Consider a hypothetical example.

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

There are \(3\) ticketbooths for selling the tickets and each ticketbooth will sell exactly \(4\) tickets.

Suppose the researcher labels the \(12\) people with ID \(1,2,\ldots, 12\) according to their arrivals.

Say, the first arrival buys his/her ticket from one of the \(3\) ticketbooths.

Then, the second arrival buys his/her ticket from one of the \(3\) 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 \(12\) people can buy tickets from the \(3\) ticketbooths?

#My Attempt:

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

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 \(12\) tickets for a documentary film and exactly \(12\) people to buy the tickets (one person can buy only one ticket).

There are \(3\) ticketbooths for selling the tickets and each ticketbooth will sell exactly \(4\) tickets.

Suppose the researcher labels the \(12\) people with ID \(1,2,\ldots, 12\) according to their arrivals.

Say, the first arrival buys his/her ticket from one of the \(3\) ticketbooths.

Then, the second arrival buys his/her ticket from one of the \(3\) 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 \(12\) people can buy tickets from the \(3\) ticketbooths?

#My Attempt:

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

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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