Okay, let's see. There's this problem about six witches and ten sorcerers at an arcane mixer. The witches shake hands with each of the sorcerers, but none of the witches shake hands with each other because they're good friends. The sorcerers don't shake hands with other sorcerers either because they hate each other. The question is asking how many handshakes took place in total. Hmm, so I need to figure out the total number of handshakes between the two groups.

First, let me break this down. There are two groups: witches and sorcerers. The key here is that handshakes only occur between a witch and a sorcerer. So, there's no handshake between two witches or between two sorcerers. Therefore, the total number of handshakes should be the number of interactions between the two groups.

Since each witch is shaking hands with each sorcerer, this sounds like a classic combinatorial problem where we have to calculate the number of possible handshakes between two distinct groups. Let me recall the formula for combinations. If there are m items of one kind and n items of another kind, the number of ways to pair them is m multiplied by n. Because each of the m items can pair with each of the n items once.

In this case, the number of witches is 6 and the number of sorcerers is 10. So, the total number of handshakes should be 6 multiplied by 10. Let me check that. If each of the 6 witches shakes hands with each of the 10 sorcerers, then for each witch, there are 10 handshakes. So, 6 times 10 gives 60. That seems right.

Wait a second, let me make sure I'm not missing anything here. The problem mentions that the sorcerers don't shake hands with each other. So, no handshakes among sorcerers. The witches don't shake hands with each other either. So, the only handshakes are between a witch and a sorcerer. Therefore, the total number is indeed the product of the two numbers.

Alternatively, if I think of it as a bipartite graph where one set is the witches and the other set is the sorcerers, the number of edges (which represent handshakes here) is the product of the sizes of the two sets. So, that's 6*10=60. Yep, that makes sense.

Is there any other angle to consider? Let's see. Suppose we tried to calculate the total number of possible interactions if everyone could shake hands and then subtract the ones that didn't happen. But that might be more complicated. Let's try that approach just to verify.

Total possible handshakes if everyone shook hands with everyone else would be the combination of 16 people taken 2 at a time, which is C(16,2) = (16*15)/2 = 120. But that's the total if everyone shook hands with everyone. However, in this problem, none of the witches shake hands with each other, and none of the sorcerers shake hands with each other. So, we need to subtract those handshakes that didn't happen.

First, subtract the handshakes among the witches: C(6,2) = (6*5)/2 = 15. Then, subtract the handshakes among the sorcerers: C(10,2) = (10*9)/2 = 45. So, total handshakes would be 120 - 15 - 45 = 60. That matches the previous result. So, that's a good check. Therefore, the answer is 60.

Wait, but let me make sure that approach is correct. The total possible handshakes without any restrictions are indeed C(16,2)=120. Then, we subtract the intra-group handshakes (within witches and within sorcerers) which are 15 and 45 respectively. So, 120-15-45=60. Yes, that's correct. Therefore, both methods give the same answer, so I can be confident that 60 is the right number.

Alternatively, another way to think about it is that each of the 6 witches shakes hands with all 10 sorcerers. So, for each witch, that's 10 handshakes. 6*10=60. Since there's no double-counting here because each handshake is between one witch and one sorcerer, there's no overlap in the counts. Therefore, this approach also gives 60.

So, all three methods—direct multiplication, subtracting from the total possible, and considering each individual's interactions—lead to the same answer. Therefore, I think 60 is the correct number of handshakes that took place at the mixer.
Therefore, the total number of handshakes that took place at the mixer is $\boxed{60}$.
</think>