- #1
songoku
- 2,328
- 335
- 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