n different toys can be distributed among n children total no of ways in which it can be distributed so that exactly one child gets no toy
One first chooses an unlucky child (n ways) and then distributes n toys between the rest n - 1 children. The latter number is (see Wikipedia). Indeed, to distribute n toys between n - 1 children so that every child gets at least one toy means to select a child that gets two toys (n - 1 ways) and then give the rest of the children one toy.
The number of onto functions(surjections) from a set of M to a set of N, , is .
Now there are n ways to select the unlucky child. There are then ways to give the toys to give the toys out so that exactly one child gets no toy.
There are choices for the unlucky child.
The six toys are distributed among the five Others.
. . Each of them get at least one toy.
Give one toy to each of the five Others.
. . There are: . ways.
The sixth toy can be given to any of the five Others: . ways.
Therefore, there are: . ways.
First select the unlucky child. This can be done in n ways.
Next distribute the first n-1 toys among the remaining children. This can be done in (n-1)! ways.
Finally, select a lucky child to get the nth toy. This can be done in n-1 ways.
So the total number of possibilities is n (n-1)! (n-1).
I suppose that Plato's answer can be simplified to this form, but just thinking about it gives me a headache.
Take the example of three toys and three children.
There are six ways to give three toys to two children so that both get at least one toy. There are three ways to have an unlucky child. That is 18 not
Soroban's count is twice the actual.
i now see my error-- I was failing to count some of the distributions like 1-1-2 in the case of 3 children, where the first two gifts do not go to the first two children.
So, in a feeble attempt to rehabilitate my reputation, here is another try at it.
First choose the unlucky child who gets no gift. This can be done in ways.
Somebody must get two gifts. These two gifts can be selected in ways.
If we consider the bundle of two gifts to be one object and the single gifts to be objects, there are a total of objects to distribute among the children. This can be done in ways.
So the total number of possibilities is .
I think this formula will be found to agree with Plato's numbers, but I still get faint at the thought of showing the formulas are the same algebraically (although I see Plato has listed them as equivalent in his post).