- #1

songoku

- 2,328

- 332

- Homework Statement
- 1) How many different ways are there to partition n persons into at most r distinct (may be empty) groups?

2) k students can choose (randomly) for themselves which one of the n tutorial groups they want to attend. What is the probability that all tutorial groups have at least one student?

- Relevant Equations
- Probability

Permutation and Combination

1) At first my answer was ##n!

\begin{pmatrix}

n+r-1 \\

r - 1

\end{pmatrix}

##

But I think that's not correct because let say first group consists of person A and B, by multiplying with n!, I also consider first group to be B and A which is just the same as A and B so there is double counting.

So I am thinking maybe ##

\begin{pmatrix}

~ & ~ n \\

n_1, & n_2, & ... & , n_r

\end{pmatrix}

##

is the answer since it is actually dividing n persons into r groups and eliminating the order inside each group. Is that correct?

2) I feel the question is similar to (1). My answer was

$$\frac

{\begin{pmatrix}

n -1 \\

k - 1

\end{pmatrix}}

{

\begin{pmatrix}

n +k -1 \\

k - 1

\end{pmatrix}

}

$$

But I think this has not considered the case where group 1 consists of person A or group 1 consists of person B. I don't think multiplying with n! works. How to get the correct answer?

Thanks

\begin{pmatrix}

n+r-1 \\

r - 1

\end{pmatrix}

##

But I think that's not correct because let say first group consists of person A and B, by multiplying with n!, I also consider first group to be B and A which is just the same as A and B so there is double counting.

So I am thinking maybe ##

\begin{pmatrix}

~ & ~ n \\

n_1, & n_2, & ... & , n_r

\end{pmatrix}

##

is the answer since it is actually dividing n persons into r groups and eliminating the order inside each group. Is that correct?

2) I feel the question is similar to (1). My answer was

$$\frac

{\begin{pmatrix}

n -1 \\

k - 1

\end{pmatrix}}

{

\begin{pmatrix}

n +k -1 \\

k - 1

\end{pmatrix}

}

$$

But I think this has not considered the case where group 1 consists of person A or group 1 consists of person B. I don't think multiplying with n! works. How to get the correct answer?

Thanks