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### De-arrangement in permutation and combination [duplicate]

This article talks about de-arrangement in permutation combination. Funda 1: De-arrangement If $n$ distinct items are arranged in a row, then the number of ways they can be rearranged such ...
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### Derangement problems [duplicate]

d(1)=0,d(2)=1,d(3)=2,d(4)=9,d(5)=44 Verify that d(5) = 44 and thus that the probability of a random rearrangement of 5 objects being a derangement is 44/120 = 0:3666 So i've been trying google/...
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### Number of permutations without constants [duplicate]

Possible Duplicate: I have a problem understanding the proof of Rencontres numbers (Derangements) Given a vector of n elements, how can I calculate the number of "true" permutations, i.e. ...
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### Number of permutations $\sigma\in S_n$ with $\sigma(k)\neq k$ for all $k=1,\ldots,n$ [duplicate]

For the symmetry group $S_n$ ($n\geq1$), how many permutations $\sigma$ exist with the property that $\sigma$ doesn't map any element of $\{1,\ldots,n\}$ to itself? I know I can try to do a counting ...
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### A question on counting and probability: Six friends hold a Christmas present swap. Each person brings a present, and puts it into a sack… [duplicate]

Question: Six friends hold a Christmas present swap. Each person brings a present, and puts it into a sack. Once all six presents are in the sack, each participant in turn then draws out a present at ...
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### Show that : $\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2= 2 n!$

I came across this result while doing some representation theory of the permutation group $S_n$ $$\sum\limits_{\sigma \in S_n} (\mbox{number of fixed points of } \sigma)^2 = 2 n!$$ This can be ...
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### Exponential Generating Function For Derangements

I have been introduced to the concept of exponential generating functions a few days ago. However, my understanding of them are still quite limited, and I would like to see some examples. Earlier this ...
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### Combinatorial argument to prove the recurrence relation for number of derangements

Give a combinatorial argument to prove that the number of derangements satisfies the following relation: $$d_n = (n − 1)(d_{n−1} + d_{n−2})$$ for $n \geq 2$. I am able to prove this ...
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### $\frac{1}{e}=$“Probability that every chocolate goes into a wrong spot”.

While watching a video by Po Shen Loh I found something strange.In the video, He said that: Suppose I have a box of chocolates having $100$ chocolates, and I drop them all on the ground, and then I ...
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### Derangements with extra chairs

This was a question on my combinatorics final. Suppose $m$ people are sitting in a room with $n$ chairs. If everyone leaves and comes back, how many ways can they sit down such that no one gets ...
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4 people are exchanging gifts. How many combinations are there so that no one receives their own gift? I tried this problem myself, and got 3!. My friends told me that it's 9. I got 3! because I ...

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