All Questions
94
questions
3
votes
3
answers
65
views
Prove that $\frac{(n + 1)!}{((n + 1) - r)!} = r \sum_{i=r - 1}^{n} \frac{i!}{(i - (r - 1))!}$
I Need Help proving That
$$\frac{(n + 1)!}{(n - r + 1)!} = r \cdot \sum_{i=r - 1}^{n} \frac{i!}{(i - r + 1)!}$$
Or in terms of Combinatorics functions:
$P_{r}^{n+1} = r \cdot \sum_{i = r-1}^{n} {P_{r-...
- 179
2
votes
1
answer
93
views
Why does the number of permutations of these sequences with non-negative partial sums have such a simple closed form (m choose n)?
I've been thinking about a problem, and I think that I have a solution, and I'm not sure why it works. Looking for an intuitive (or just any) explanation.
The problem
Choose an integer $k>1$. For ...
- 61
1
vote
1
answer
43
views
Summing a binomial series that arose while counting functions
Define $f:A \to A$ where $A$ contains $n$ distinct elements. How many functions exist such that $ \forall x \in A, f^m(x)=x$, $(m<n)$ (and $m$ is prime to avoid the mistake pointed out in the ...
- 739
2
votes
2
answers
85
views
$n$ chips for 100 cookies problem — is there a counting solution?
The problem is "You are making 100 cookies. How many chips $n$ do you need to put into the batter to have at least 90% probability that every cookie has at least one chip?"
I tried a stars ...
1
vote
1
answer
71
views
Why does ${}^4 C_2 \times {}^8 C_3$ not give the number of $5$ member committees (with at least $2$ women) from $4$ women and $6$ men?
The question is find how many ways we may select a committee of $5$ members from $6$ men and $4$ women such that at least two women are included.
I know that the standard approach to this is to take ...
0
votes
1
answer
90
views
Probability / Permuations: Expected Number of Games Till Bust
You bet 1 dollar in a game in which the win probability of each round is 0.55. As long as you don't go bust (have $0 left), you could bet up to 100 times. You start with 4 dollars in the bank. What is ...
- 3
3
votes
2
answers
894
views
Permutation and Combinations to select a committee and election board
I am currently studying for the SOA exams. I have ran into a problem in Lecture Notes in Actuarial Mathematics - A Probability Course for the Actuaries - Marcel B. Finan that I thought was quite ...
0
votes
0
answers
98
views
Get mixed sum-constrained and unconstrained combination sequence by its index
This is a harder variation of this problem. We are given a combination formed by sum-constrained and unconstrained parts:
n - sum-constrained sequence elements sum,
k1 - length of sum-constrained ...
0
votes
1
answer
65
views
Get the sum-constrained combination sequence by its lexicographical index
I need to solve the inverse of this problem. Suppose we are given the rank 49510, meaning we want to find the 49510-th ...
3
votes
4
answers
178
views
What is the flaw in this approach?
$12$ delegates exists in three cities $C_1,C_2,C_3$ each city having $4$ delegates. A committee of six members is to be formed from these $12$ such that at least one member should be there from each ...
- 10.2k
3
votes
2
answers
186
views
Question from isi previous years
(a) Show that $\left(\begin{array}{l}n \\ k\end{array}\right)=\sum_{m=k}^{n}\left(\begin{array}{c}m-1 \\ k-1\end{array}\right)$.
(b) Prove that
$$
\left(\begin{array}{l}
n \\
1
\end{array}\right)-\...
- 563
1
vote
1
answer
174
views
Counting the number of dominating rook placements in a chessboard
Given a square $n \times n$ chessboard and $m$ rooks (with $m \geq \lceil{n/2}\rceil$ and $m \leq n^2$) I would like to count how many of the total $\tbinom{n^2}{m}$ possible combinations cover each &...
- 155
1
vote
1
answer
66
views
Evaluating the sum over all strings made of two anticommuting terms
Given two anticommuting elements, $A$ and $B$, I aim at evaluating the sum over all strings of length $n$ multiplying exactly $k$ elements $A$ and $n-k$ elements $B$ (as we know, there are $\binom{n}{...
- 115
0
votes
1
answer
317
views
A permutation approach of Lilavati Book
Here is the Bhaskaracharya's Lilavati book by translated John Taylor,1816.
https://books.google.co.in/books?id=0KMIAAAAQAAJ&printsec=frontcover&hl=tr#v=onepage&q&f=false
I am looking ...
- 421
0
votes
1
answer
242
views
Expected number of cycles in random permutations
Draw at random a permutation $\pi$ in the set of permutations of $n$ elements, $S_n$, with probability,
$$
P(\pi)= \frac{N^{L(\pi)}}{ \sum_{\pi \in S_n} N^{L(\pi)} },
$$
where $ L(\pi)$ is the number ...
- 200