Probability of knives and forks matching 
Six knives and six forks come in pairs. Two pairs of each are red, two are white and two are black. If the knives are matched randomly with the forks, what is the probability that no pairing has the same colour?

I tried to do this the long way and multiply the probability of no match for each successive pairing, but I'm not sure how to take into account all of the different combinations. Could someone please direct me, I'm not great at probability problems.
 A: There are $6!=720$ possible assignments of knives to forks.  Let's figure out how many of them are "good."
There are really only two cases to consider.  First, let's consider the placement of the red knives.  For the matching to be "good,"  either both red knives must pair with two knives of the same non-red color, or one red knife must match with white and the other must match with black.
There are $4$ "good" ways for red knives to match with the same color -- once $R_1$ chooses any of the four permissible forks, we know where $R_2$ has to go, and $8$ "good" ways for red knives to match with different colors.
Let's assume that both red knives are paired with black forks.  Then both black knives must pair with white forks ($2$ possibilities) and both white knives must pair with red forks (another $2$ possibilities).  That means there are $16$ "good" pairings of this form.
Now let's assume that one red knife is paired with a black fork and the other red knife is paired with a white fork.  Recall that this can happen in $8$ different ways.  Then one of the black knives must pair with the remaining white fork ($2$ possibilities because it could be either of the black knives) and the other black knife must pair with one of the red forks ($2$ possibilities).  There are then $2$ possibilities for the white forks.  That means a total of $64$ "good" pairings of this form.
Thus, there are $80$ "good" pairings out of $720$ possible pairings, so the probability of a "good" pairing is exactly $\frac 19$.
