Questions tagged [combinatorics]
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.
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I tried this question by using pi denoting method and got a big equation what to do
Find coefficient of $x^6$ in
$(1+x)(1+x^2)^2.....(1+x^n)^n$
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1
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From the mountain top to the coast.
From a mountain top lead to ways to the (sea)coast. Neither way goes below of the sea level or over of the mountain top. Show that Adam and Barbara can go on the roads from the mountain top to the sea ...
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How many bit strings of length 20 have exactly five $1$’s and do not contain $11111$ as a substring? [closed]
Recall that a bit string is a string composed of characters $0$ and $1$.
Can someone explain how the answer is:
${20\choose5} - 16$?
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help me in solving [closed]
evaluate :
$$
\sum_{k=0}^\infty\binom{n}{2+4k}
$$
I tried using pre formulated series of multinomial expansions but it doesnt help. Please give a solution to the problem, without using complex ...
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game theory+probability problem [duplicate]
In one city, $N$ Petya, Vasya and Tolya hide from zombies in an underground bunker. But they have no connection with the outside world and they don’t know if the zombies remained in the city or left. ...
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Binomial and combinatorics [closed]
Please help me solve this problem.
Show that:
$$\sum_{k=1}^n k{n \choose 2k+1} = (n-2)2^{n-3}$$
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Find cofficient of $x^p$ in the expansion of $(ax^2+bx+c)^n$?
Here p can be $0, 1, 2...$ and $n$ can be $0,1,2,...$ I just want to get the generalized relation for finding the coefficient of $x^2$ or $x^3$ and so on in expansion of multinomials like $(1+x+x^2)^5$...
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No two children may sit in adjacent seats [closed]
There are twelve people which includes 3 couples, 3 single adults and 3 children. In how many ways they can be arranged :-
a) if no two children can sit in adjacent seats?
b) if each couple must sit ...
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3
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Solve algebraically $n \binom{m+n}{m} = (m+1)\binom {m+n}{m+1}$ [closed]
I can't get very far with this one :/
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partition of a multiset
Let $X=\{\underbrace{a_1,\cdots ,a_1}_{\nu_1},\cdots,\underbrace{a_k,\cdots ,a_k}_{\nu_k}\}$
be a multiset of cardinality $\sum{\nu_i}=n$ where each $a_i$ repeats $\nu_i$ times. We suppose that when $\...
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What is the number of permutations for given N numbers, such that the first part is non-decreasing?
Let $A$ be a list of $n$ numbers in range $[1,100]$ (numbers can repeat).
I'm looking for the number of permutations of $A$ which start with a non-decreasing part, where this part ends with the first ...
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2
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Some integers related to the Hilbert scheme of points in the plane. [closed]
This question is related to another question posed on this site.
Let me recall the construction: Let $A:=k[x,y]/I$ with $k$ the complex numbers (or any algebraically closed field) and $\dim_k(A)< \...
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