Permutations and Combination

We will start by considering the Xs. There must be one in the top row, so we have three places that X could be placed. If we look at the situation carefully, we can see that we can place the other two Xs two different ways for each starting X position. So, to start, we have 3 positions for the first X and 2 ways to distribute the other two Xs. So we are up to 6 possibilities for where we can place our Xs.

Now we consider the Ys. We can place the first Y in either of the unoccupied spaces on the first row. However, once we do so, there is only one way we can put the other two Ys into the grid so they don't overlap. So we can double our possibilities to 12.

Once our Xs and Ys are set, the Zs will fall into place. We have no choices as to where to put them.

So the total number of possibilities is: 3 initial spots for the X on the first row times 2 ways for the other Xs to fall out times 2 ways for the Ys to be placed. Thus, the total is (3)(2)(2) for a total of 12.