Can anyone explain what is the meaning of that leaves 8 more from SIR’s solution ?

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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.

@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.

3

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.

$$

\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.