Conditional Permutations

Question

There are four teams (A, B, C, D) participating in a swimming tournament ; each having three participants. In how many ways can the 12 participants can be made to sit in a row such that each of the team A members are sandwiched by participants from the other teams?

Answer 1

Hint

In the diagram below, the bullets represent members of the "other" teams,
and the up arrows, places where members of team A can be inserted.

You should be able to figure it out from here.

$\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet\uparrow\bullet$

Choose 3 places from 8 for A team members, permute them, permute 9"others"

Answer 2

Think of the question this way: Since there are four teams and each team has three members i.e., $12$ total members then there will be $12$ seats required. Now, since Team $A$ members have to be sandwitched by other team members so, on the first seat, we can choose a member to seat out of $12-3=9$ (i.e., Total - Team $A$ members). That means on the first seat 1 out of $9$ members can sit. Then on the second seat there will be $1$ out of $8$ (but depends, may be you want Team A members to sit on second seat). Now do this for every seat until you get your answer!