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Mathematics

Questions tagged [combinatorics]

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For questions about the study of finite or countable discrete structures, especially 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.

71 questions from the last 30 days
Newest Active Bountied 3 Unanswered
11 votes
4 answers
439 views

Evaluate: $$4^9-\binom{8}{1}4^8+\binom{7}{2}4^7-\binom{6}{3}4^6+\binom{5}{4}4^5$$ My Attempt Doing normal arithmetic calculation the value comes out to be $5120$ which is not a big deal. But can it be ...
Maverick's user avatar
  • 11.2k
8 votes
2 answers
760 views

Let $n\in\mathbb{N}$ and $X_1,\dots,X_4$ be independent and uniformly distributed on $\{1,\dots,n\}$. What is the probability that the tuple $(X_1,X_2,X_3,X_4)$ can be partitioned into two (multi)sets ...
johnny10's user avatar
  • 307
7 votes
3 answers
257 views

I am trying to solve the following problem on integer sequences and subset sums from a 2023 Shanghai high school entrance exam: Let $A = (a_1, a_2, \dots, a_n)$ be a sequence of positive integers ...
thedeepdeepsky's user avatar
  • 1
3 votes
2 answers
404 views

Given a graph, I wish to try and color it with the minimum number of colors (0,1...k). Each 2 connected vertices must have different colors. I proposed the following algorithm - Initialize an array of ...
C. Arnold's user avatar
  • 33
5 votes
2 answers
598 views

Problem statement: If $\frac{n!}{3!(n-3)!}:\frac{n!}{5!(n-7)!}=1:6$. Find $n$. This simple factorial equation appeared in my test today. It was an integer type problem, which means only one integer ...
div_24's user avatar
  • 171
7 votes
2 answers
356 views

There are $𝑛$ consecutive integers $𝑚,𝑚+1,\dots,𝑚+𝑛−1$. Prove that you can choose some nonempty subset of these numbers whose sum is divisible by $1+2+\dots+𝑛 $. Edit: Okay, so here is what I ...
Obtuse's user avatar
  • 109
3 votes
3 answers
346 views

Q: How many ways to arrange BOOKKEEPER where two E’s appear consecutively but not three. Here What I've got : a) We can consider the two consecutive E’s as one block say X. Hence, we get a new string: ...
Jonathan's user avatar
  • 33
1 vote
2 answers
110 views

Find the number of quadruples of non negative integers (a,b,c,d) such that $105a + 70b + 42c + 30d = 2025$ Simply using generating functions makes things very lengthy: This problem is similar to ...
Cuckoo Beats's user avatar
  • 579
7 votes
1 answer
291 views

Consider natural numbers $e_1 < e_2 < ... < e_k <n$. Define the graph $G$ to have the vertex set $V= \{1,\dots, n\}$ and edge set $E = \{(a,b)\in V^2 \: |\: \exists j=1,\dots, k: a = b\pm ...
Joseph Expo's user avatar
  • 668
3 votes
2 answers
148 views

Let $E$ be a finite set of points and $\mathcal{F}$ be a collection of subsets of $E$ (lines) such that each line has cardinal at least 4 two distinct lines intersects at exactly one point. Show ...
Pii_jhi's user avatar
  • 79
5 votes
1 answer
189 views

Question Consider a linear arrangement of $10$ balls selected from an infinite supply of blue and red balls. Determine the total number of distinct arrangements that satisfy the following condition: ...
thedeepdeepsky's user avatar
  • 1
5 votes
2 answers
197 views

I am trying to find a closed form for the following sum: $$ \sum_{k=0}^{n-1} \left( \frac{1}{(k+1)(n-k)} \cdot \binom{n+1}{k+1}^2 \right) $$ What I have tried so far I tried to simplify the expression ...
Alex's user avatar
  • 51
0 votes
2 answers
191 views

If there are 8 letters numbered 1 1 2 3 4 5 6 7 (not 12345678) instead there is one 1 extra and similarly there are envlopes numbered 1 1 2 3 4 5 6 7 assuming that the two letters and the two ...
Marvelmaanas12's user avatar
  • 23
2 votes
1 answer
227 views

In the game "Tug of Luck" $n$ coins are tossed. Player A gets the tails and B gets the heads. Thereafter they take turns rolling a die until one player has gotten rid of all their coins and ...
Rüdi Jehn's user avatar
  • 644
1 vote
2 answers
127 views

Comment. I found this solution but I do not understand the identity in the first centered formula. It looks similar to something from probability theory, like the expected value of a binomial random ...
user217519's user avatar
  • 520

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