# 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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### Stirling number of Second kind generating function

I would like to prove that: $$f_m(x) = \dfrac{x^m}{(1-x)(1-2x)...(1-mx)}$$ Where $$f_m(x) = \sum_{n=0}^{\infty} S(n,m)x^n$$ and $S(n,m)$ is stirling number of 2nd kind Multiplying the recurrence ...
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### Probability that the tallest and shortest person are sitting next to each other if they cannot sit at either end?

Eight people of different heights are to be seated in a row. The shortest and tallest in this group are not seated at either end. What is the probability that: (a) The tallest and shortest ...
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### number of ways to stack n distinct objects into k distinct boxes

I know that number of ways to distribute $n$ distinct objects into $k$ distinct boxes is $k^n$, but there order of objects in a box doesn't matter. If we want to stack objects in a box, then order ...
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### Number of ways in which 6 rings can be worn on the 4 fingers of one hand

The way I solved this is - The 1st finger can have any of the 6 rings, $\therefore 6$ ways The 2nd finger can have any of the 5 remaining rings, $\therefore 5$ ways The 3rd finger can have any of ...
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### Combinatorial problem : How many ways can you choose 3 balls from nine such that no two of them are consecutive?

I tried approaching this problem by first assuming that all the balls are chosen from the odd places(10) only and then the even places only(6). And then adding them together. However, my approach ...
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### Double Summation Over all subset of $\{1,2,…n\}$

In Benson's Book "Polynomial In variants of Finite Groups" It is claimed that(Without any proof): $$j! u_1u_2...u_j =\sum_{I \subseteq \{1,2,...,j\} } (-1)^I (\sum_{i \in I}u_i)^j$$ Where $I$ runs ...
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### Arrangement of people in 2 taxis.

Find the number of ways in which 6 people can be seated in 2 taxis of 4 seats each if internal arrangement matters. My answer is $\frac{6!6!}{4!2!}$ But the answer key says different. Please help
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### The Basic Principle

In any n+1 integers there will be a pair which differs by a multiple of n. I have tried to create a pigeon hole with numbers a0,a1,a2,...,an but i could not get a solution.
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### The number of positive integral solutions of the equation $x_1x_2x_3x_4x_5=1050$? [duplicate]
The number of positive integral solutions of the equation $x_1x_2x_3x_4x_5=1050$ ? I prime factorized 1050.Then what to do?
In a game of tickets,the tickets are marked $1,2,3,...,50$.If 6 unordered combination of tickets is selected,then the probability that the subset will have at least one neighbor present is [if at ...