in Combinatory recategorized by
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3 votes
3 votes
We go to a pizza party, and there are $5$ types of pizza. We have been starving for days, so we can eat $13$ slices, but we want to sample each type at least once. In how many ways can we do this? Order does not matter.
in Combinatory recategorized by
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3 Comments

Can anyone explain what is the meaning of that leaves 8 more from SIR’s solution ?
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@Amar123 Total there is 13 slice we can eat which consist of 5 types of pizza .

Each slice is from any of the 5 type of pizza.

So it is given we have to include all types of pizza at least one .

So from 13 slice 5 slices are fixed as 5 types of pizza .

there are 8 slices possible which can be fill be any of 5 types of pizza.

P1+P2+P3+P4+P5=8 

now apply star and bar and solve.

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Yeah understood !! Thanks @Kabir5454

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1 Answer

5 votes
5 votes
First, we sample the $5$ types. That leaves space for $8$ more, which we can choose freely, with repetition. The dots-and-bars(stars and bars) formula tells us that there are
$$
\left(\begin{array}{c}
8+5-1 \\
5-1
\end{array}\right)=\left(\begin{array}{c}
12 \\
4
\end{array}\right)
$$
ways to do this.
Answer:

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