I was doing a seemingly trivial question, and I though it was a simple application of the counting theorem but it turns out it doesn't work. Here's the question
From a deck of 52 cards, how many ways are there to arrange a hand of 5 cards such that all 4 kings are in the hand (order doesn't matter) (the last card can be any non-king)
Now here's my thought process as an application of the counting principle: $$ \frac{4 \times 3 \times 2 \times 1 \times 48}{5!}$$
As we have $4!$ ways of placing the kings and then the last card can be from 48 other cards. Then we divide by $5!$ to remove the order. Unfortunately, this produces a non-integer so I was very sad indeed. However, it logically seems like it should work as it follows what I think is valid logic. Could someone explain how to get the correct answer (48) and also more importantly, why my logic was incorrect?
When you write $$4\times 3\times 2\times 1\times 48$$ you haven't chosen yet exactly where the non-king is in the sequence.
Yet when you divide by $5!$ you pretend that you have done so. That's where your argument fails.