I've stumbled upon the question of:
Six diplomats are awaiting assignments to foreign embassies. If 1 of these is to be assigned to Germany, 2 to France and 3 to England, in how many ways can the arrangement be made?
I thought that it could be: p(6,1)+p(5,2)+p(3,3) However That seems quite off now that I look at it
Let's go over this question.
So we have six people.
We pick any one of these in six ways, to go to Germany.
Then, we pick any two of them, to go to France. This is done in $\binom{5}{2}$ ways.
The remaining three will all go to England, so there is no choice there.
Hence, all we have to do is to find the number of ways of sending one person to Germany and two to France. This is $6 \times \binom{5}{2} = 60$ ways.
A better answer is that this is just $\binom{6}{3,2,1}$, the multinomial expression, which evaluates to $\frac{6!}{3!2!1!} = 60$.