Factorial with names

Question

Ok so, I have had an argument with my teacher over 1 quiz question that was marked wrong in my data management class.

Question. Determine the number of ways that 12 members of the boys' baseball team can be lined up if Joe, Tanner, Josh must all be together.

I wrote the answer as 10! x 3 so 10,886,400.

My teacher had it as 10! x 3 x 2 so 21,772,800

If my answer is wrong, can someone explain to me how he got that 'x 2'?

Answer 1

The $10!$ comes from treating the three boys as one and so we are now considering 10 boys and asking how to line them up. However, in the grouping of the three boys, they can be permuted among themselves in $3! = 3 \times 2$ ways.

Answer 2

You would be right if Joe and Josh were identical twins. One way to think about it is you have a block of $3$ boys who can be placed in any of the $10$ spaces between or to either side of the other $9$, so the answer is $3! \cdot 9! \cdot 10$.