Distributing items of different varieties

Find the number of ways to fill a box with a dozen doughnuts chosen from five

varieties, with the requirement that at least one doughnut of each variety is

picked.

Here is how I think this is done, is this correct?

1. Put one donut of each type in the box. 7 Donuts now left.

2. These donuts can be divided using the | and * method. So *******|||| stands for 7 donuts and 4 separators for the varieties, 11 items in total. The dividers can be placed in 11 Choose 4 = 330 ways.

Is this correct?

Re: Distributing items of different varieties

Quote:

Originally Posted by

**nukenuts** Find the number of ways to fill a box with a dozen doughnuts chosen from five varieties, with the requirement that at least one doughnut of each variety is picked.

The best way to do these is to learn the multi-selection formula.

The number of ways to put *N* identical items into *K* different cells is $\displaystyle \binom{N+K-1}{N}$.

That is proven using the "stars & bars" as you suggested.

In your problem the identical items are the choices and the different cells are the varieties of doughnuts. But we have the requirement that no cell be empty. So put one of each variety in the box. Now count the number of ways to make an addition seven choices: $\displaystyle N=7~\&~K=5$.

Re: Distributing items of different varieties

Cheers mate, good to know that there is a standard formula for this process.