Distributing $10$ unique candies to $2$ kids - two models We have got $10$ unique types of candies and two kids. We need to distribute all of the candies to them, making sure that each of them gets at least one candy. I have come up with two models to solve this problem but can't decide which of them is correct and why.


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*We distribute all of the candies at random and then subtract the two events when one of the kids has all of the candies : $2^{10}-2$ possible ways

*We choose one out of $10$ candies to give to the first child and then one of the $9$ remaining candies to give it to the second one. Then, we distribute all of the remaining candies at random, which gives $10 \times 9 \times 2^{8}$ possible ways. 

 A: By the 2nd method, if you start by freely choosing one of the ten candies - lets say you choose the stick of rock, and then distribute the other 9 subsequently, the 2nd method you describe counts that as a different distribution to the case in which you first choose something else, and the stick of rock is the 2nd candy distributed.
So the 2nd method would be correct if each child ate the sweets in the order they were distributed and you wanted to know how many ways there were of eating the sweets.
The first method is correct.
A: Quite simply, as lulu said, do it for a set of smaller candies. 
For example, two candies between two people. Obviously, each person gets one candy, leaving one combination, as the other two combinations are not valid for this set of criteria. As for the layout of the candies, you can have $C_1$/$C_2$ or $C_2$/$C_1$, leaving two ways. (You are right as normally, you can put it in $2^2 $ ways, but removing the other combination, you get $2^2-2$ or in other words, $2$)
Three candies between two. You can have: 


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*$P_1$ have one, $P_2$ have two

*$P_1$ have two, $P_2$ have one
There are only three ways to arrange three sweets in a [1,2] format, as $C_1 / C_2, C_3$ is the same as $C_1/C_3,C_2$. $3*2 = 6$. Also, this can be obtained by $2^3-2$


Four candies between two. You can have: $[1,3],[2,2],[3,1]$
etc. 
Can you see the pattern?
Carry this pattern for ten candies, and you get: (DON'T HOVER OVER IT IF YOU DON'T WANT THE ANSWER, IT IS A SPOILER!)

 $2^{10} - 2 = 1024 - 2 = 1022.$ We have got the pattern that to share $n$ candies between two people, there are $2^n - 2$ combinations! (If this is a homework question, try with other values! It would help! 

