how many ways can ymir, bin, charles, and daria stand in a line if ymir and bin refuse to stand next to each other?
We can use completmentary counting here. Let Y stand for Ymir and B stand for Bin and C stand for Charles and D stand for Daria. Then, we get that:
There are a total of 4! ways in order to rearrange this letter. However, let's find the opposite of what we want. We can group the term Y and B together and form a new letter, which we can call X. Hence, we get:
There are 3! ways for which we can rearrange XCD, however, we have to multiply this by 2 since X can be rearranged in 2! ways. Hence, we get 3! * 2 = 6 * 2 = 12.
We can now subtract this from 4! = 24, and get 12.