All Questions
181
questions
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38
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Understanding the Derivation of a Formula Involving Binomial Coefficients and Factorials
I'm studying a formula that involves binomial coefficients and factorials, and I'm struggling to understand how it was derived.
The image below is a screenshot from the paper. They are taking the ...
- 326
2
votes
2
answers
74
views
Correctness of Solution for Forming a Committee with More Democrats than Republicans
I recently encountered a problem and derived a solution, but I am uncertain about its correctness. Here's the problem:
At a congressional hearing, there are 2n members present. Exactly
n of them are ...
- 33
6
votes
1
answer
147
views
How to find all positive integers $n,k$ such that ${n\choose k}=m$ for a given $m$?
This question is motivated by a simple exercise in Peter Cameron's Combinatorics: Topics, Techniques, Algorithms:
A restaurant near Vancouver offered Dutch pancakes with ‘a thousand and
one ...
- 2,454
4
votes
2
answers
187
views
Combinations of indistinguishable marbles
Let's consider this problem:
A bag contains 5 black marbles and 6 white ones. Marbles of the same color are indistinguishable from each other. If I draw two marbles, what is the probability they have ...
- 43
0
votes
1
answer
49
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Can we solve this only by using bijection?
The question is :
The number of possible outcomes in a throw of n ordinary dice in which at least once of the dice shows an odd number are:
Now, we can simply apply bijection principle , and ...
- 149
0
votes
0
answers
26
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Choosing k positions among n with no more distance d each other
Let $\binom{n}{k}_{<d}$ the number of combinations of k elements among [1,n] with constrained spacing :
no element can be at distance d or more from its successor.
$$\binom{n}{k}_{<d} = \sum_j (-...
- 185
0
votes
0
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61
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Why is $\binom{n}{1}\binom{n-1}{1} \neq \binom{n}{2}$? [duplicate]
$$
\binom{n}{1}\cdot\binom{n-1}{1} = n \cdot (n-1),
\qquad\text{whereas}\qquad
\binom{n}{2} = \frac{n(n-1)}{2}. $$
It's quite obvious that LHS $\ne$ RHS.
But intuitively we know, as per the definition ...
- 331
1
vote
2
answers
85
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Prove that $\sum\limits_{i=0}^n(-1)^i\binom{n}{i}\binom{kn-ki}{n-1}=0$
Given $$\sum\limits_{i=0}^n(-1)^i \binom{n}{i}\binom{m-ik+n-1}{n-1}$$ (which can be interpreted as the number of solution sets to the equation $x_1+x_2+\cdots+x_n=m$ where $0≤x_i<k$), prove that $$\...
- 637
0
votes
1
answer
70
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A committee of 11 members is to be formed from 8 males and 5 females with at least 6 males in the committee [duplicate]
Q) A committee of 11 members is to be formed from 8 males and 5 females. The number of ways the committee is formed with at least 6 males is?
The general solution I see everywhere is to take 3 cases ...
- 1
0
votes
4
answers
58
views
Solving Combinatorial Problem - Red Ball Distribution in Three Bins
I'm reaching out to the community for help in tackling a combinatorial problem that I'm currently stuck on. The problem revolves around distributing twelve identical red balls into three bins, labeled ...
- 379
2
votes
1
answer
62
views
Simplifying the sum $\sum_{ \substack{i,j,k\geq 0;\\ i+j+2k=r}} (-1)^i {n\choose i}{n \choose j}{n \choose k}$
We assume that $n,r\in \mathbb{N}$. I've been pondering on this on and off for a few weeks now. Negating the index with $i$ doesn't seem to help.
$$
\sum_{ \substack{i,j,k\geq 0;\\ i+j+2k=r}} {i - 1- ...
0
votes
1
answer
82
views
Number of combinations of size $k$ of elements in a multiset with finite multiplicites
Suppose I have 3 types of balls each in varying amounts, e.g. 3 Red (R) balls, 3 Green (G) balls, and 2 Blue (B) balls. How many combinations of size $k, k\leq n$ where $n$ is the total number of ...
- 83
1
vote
2
answers
214
views
Combinations with repetition? $n+r-1 \choose r-1$=$n+r-1 \choose r$
There is one section in my book called "Combination with repetitions".
The proof there is done with bijection. I wasn't able to understand the proof. I went to youtube.
In one video the ...
2
votes
2
answers
157
views
Is $\sum_{s=n}^{m}\binom{m}{s}(x-1)^{m-s}=\sum_{s=n}^m\sum_{j=s}^m\binom{s-1}{n-1}\binom{m-s}{j-s}(x-1)^{m-j+s-n}$ an identity?
Is there any chance that this:
$$\sum_{s=n}^{m}\binom{m}{s}(x-1)^{m-s} = \sum_{s = n}^{m} \sum_{j=s}^{m}\binom{s - 1}{n - 1}\binom{m-s}{j-s} (x- 1)^{m - j +s - n}$$
is an identity? If so, what is its ...
- 1,704
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 ...
