Tagged Questions

For questions about the study of finite or countable discrete structures, specifically how to count or enumerate elements in a set (perhaps of all possibilities) or any subset. It includes questions on permutations, combinations, bijective proofs, and generating functions.

147 views

Summation with combinations

Prove that $n$ divides $$\sum_{d \mid \gcd(n,k)} \mu(d) \binom{n/d}{k/d}$$ for every natural number $n$ and for every $k$ where $1 \leq k \leq n.$ Note: $\mu(n)$ denotes the Möbius function. I have ...
73 views

Supposedly really hard problem involving combinations

This problem gives 7 (max) out of 100 points for a college entrance exams. Seems odd because it looks easy to me, although my combinations are not too good. There are $10$ people forming a ...
32 views

Degree of Jacobian of homogeneous polynomials

What is the degree of the Jacobian (as a polynomial) of 3 homogeneous polynomials in 3 variables of degrees say $m_1, m_2$ and $m_3$ ? I don't know how to prove that it is independent. In my case the ...
55 views

Number of even numbers having digit 2 in them.

I am trying to count numbers from 1 to N which exist in A121022 but I am unable to think of solving in better than O(NLog(N)) , can you suggest a better algorithm?
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Probably an ambiguous word problem

I don't know if this should have been posted on English because it's about interpretation of a sentence, or Math because it involves with a math problem to get the right context and interpretation... ...
33 views

Combinations of sandwiches

My stats summer packet proposes the question: "if a sandwich shop has $3$ different types of meat, $4$ different types of bread, and $3$ different types of cheese. How many types of sandwiches can you ...
34 views

mutual information and combinatorics

\begin{align} &\mathrm{H}\left(\frac{1}{2^{k}}\right) \\[3mm]&\ \!\!\!\!\!\!\!\!\!\! - {1 \over 2^{k}}\left\{% {k \choose 0}\mathrm{H}\left(\left[1 - \epsilon\right]^{\,k}\right) + {k \choose ...
50 views

Is this equation true?

As the question states, does this equation hold true? $\sum_{j=0}^n \sum_{E \in {n \choose j}} (-1)^{|E|}(n-|E|)! = \sum_{j=0}^n(-1)^j(n-j)!{n \choose j}$ From what I understand, this holds true at ...
79 views

Numbers with constant digit-sum in increasing order

For base $b = 10$, I want to list all numbers with $d$ digits (no leading zeros) and digit sum $x$, in increasing order. For example for $d = 6$ and $x = 40$ we would get: 139999, 148999, 149899, ...
37 views

Probability of 4 specific numbers (1-3000) occuring in a sample of 400

How to calculate the probability that four specific, distinct numbers from the range 1 - 3000 occur at least once in a fixed sample of 400 random numbers from the range 1-3000? The numbers in the ...
16 views

Existence of Kazhdan Lusztig basis proof due to Soergel

This question is regarding the proof of the existence and uniqueness of Kazhdan Lusztig basis theorem for an arbitrary coxeter group $W$ due to Kazhdan and Lusztig in his paper "Representations of ...
26 views

Number of licence plates that match a criterion [closed]

A new license plate in Alberta consists of three letters followed by four numbers. Letters are chosen from a list of $24$ acceptable letters that may be repeated. And any digits can be used and they ...
19 views

When arranging numbers and letters in combinatorics, should one use multiplication or addition?

Let's say that we are given that a code is formed with 3 letters of alphabet followed by 3 digits from 0-9, and both can be repeated. When required to find the total number of combinations. Is it ...
33 views

stirling numbers of second kind

i am new to combinatorics and just encountered stirling numbers of second kind the book i am using does not provide much info about it except number of ways of distributing "r" distinct objects ...
sum of falling factorial $\sum_{k=0}^{n-1}\frac{n!}{(k+1)!}a^{n-k-1}$
I want to compute $\sum_{k=0}^{n-1}\frac{n!}{(k+1)!}a^{n-k-1}$. I note that it is similar to a generating function. The coefficients are falling factorials. Can I simplify it? Thanks!