# 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.

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### Simplify this equation.

Can I simplify or approximate this equation without sigma and combination? \begin{align} \sum_{i = 0}^n (-1)^i {n \choose i} \frac{{d+1}}{d(di + 1)} \end{align}
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### Is there a closed form for this binomial sum?

I am looking for a closed form of this sum:$\sum\limits_{j=k}^n\binom{j}{k}(-1)^j$ I know that this sum has a closed form: $\sum\limits_{j=k}^n\binom{j}{k}=\binom{n+1}{k+1}$ I can get this closed ...
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### in urn A white balls and B black balls. what would be the probability of taking the 5th ball being white

the problem goes like that "in urn $A$ white balls, $B$ black balls. we take out without returning 5 balls. (we assume $A,B\gt4$) what would be the probability that at the 5th ball removal, there was ...
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### How many 6-digit numbers contain exactly 4 different digits? [duplicate]

my solution is----> NUMBER can be 777210 this i denote by aaabcd ------ this can be done in ---> 10*1*1*9*8*7*[6!/3!] {1 for a thrice} NUMBER can be 772210 this i ...
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### Transforming generating functions into algorithms that generate combinatorial objects

I've stumbled upon this paper where they write about random sampling of combinatorial objects. For sampling to be proper one has to know some core numbers (probabilities). However, I'm not interested ...
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### Help me understanding what actually i counted with inclusion-exclusion

I tried to solve following task: Count number of $8$-permutations from $2$ letters $A$, $2$ letters $B$, $2$ letters $C$ and $2$ letters $D$ where exactly one pair of same letters are adjacent in ...
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### Arrangement of 12 boys and 2 girls in a row.

12 boys and 2 girls in a row are to be seated in such a way that at least 3 boys are present between the 2 girls. My result: Total number of arrangements = 14! P1 = number of ways girls can sit ...
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### When is a recurrence the sum of the powers of the roots of a polynomial?

Newton's formula allows one to calculate the sum $S_n(P)$ of the $n$th powers of the roots of a given monic polynomial $P$ without finding the roots explicitly. (This works even when the roots ...
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### Parity of $\sum_{i=1}^{n}\lfloor \log_2(i) \rfloor$

Let, $L=\sum_{i=1}^{n}\lfloor \log_2(i) \rfloor$. Problem: Find $n$ for which $L$ is odd. In other words, find a closed form expression (function) $f(n)$of variable $n$ such that $L$ is odd/even if ...
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### Find general solution for the equation $1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r)$

A positive integer $n$ is called a balancing number if $$1 + 2 + \cdots + (n − 1) = (n + 1) + (n + 2) + \cdots + (n + r) \tag{1}$$ for some positive integer $r$. Problem: Find the general ...
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### Least distance between two points in an equilateral triangle [closed]

Five points lie inside an equilateral triangle of side 2 units.Prove that at least 2 points are no more than a unit distance apart.
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### Picking balls from boxes, a logical approach?

You have a box with ten purple balls, five red balls, five blue balls, three yellow balls. You pick out four balls at random. What is the probability of all four balls being the same color? I've ...
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### Differentiating the binomial coefficient

I took a lecture in combinatorics this semester and the professor did the following step in a proof: He showed that function $f: x \mapsto \binom{x}{r}$ is convex for $x > r - 1$ (in order to use ...
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### How to find number of integral solutions, containing large number of cases?

Number of positive unequal integral solutions of the equation $x+y+z=12$ can be found out knowing the cases it involves: $(1, 2, 9) , (1,3,8), (1,4,7), (1,5,6), (2,3,7), (2,4,6) and (3,4,5)$. Thus, ...
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### Problem 14, Ch. 1 from Blitzstein and Hwang, Intro to Probability

You are ordering two pizzas. A pizza can be small, medium, large or extra large, with any combination of 8 possible toppings (getting no toppings is also allowed, as is getting all of 8). How many ...
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### Problem 13, Ch1 from Blitzstein and Hwan, Intro to Probability

A certain casino uses 10 standard decks of cards mixed together into one big deck, which we will call a superdeck. Thus, a superdeck has $52{\cdot}10=520$ cards, with 10 copies of each card. How many ...
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### Combination Problem : $6$ Countries , $4$ players from each country
$6$ Countries participate a world tournament . Each country has $4$ players. One Cricket player , One Rugby player , one Volleyball player and one Football player. Need to select a team of $8$ ...