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:

 

YBCD.

 

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:

 

XCD

 

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. 

YMIR AND BIN  NOT NEXT TO EACH OTHER

 

y x b x

b x y x

x y x b

x b x y

y x x b

b x x y       six ways       

                   for each of them the other two spots can be filled in two ways with charles and daria   6 x 2 = 12 ways (as ds1119 found)