Number of ways of shuffling cards such that $i^{th}$ card is not in the $i^{th}$ position

Asked Mar 09 '16 at 18:48

Active Mar 09 '16 at 18:51

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Suppose we have $n$ cards. How many ways of shuffling do we have such that card $i$ is not in the $i$ slot.

I thought about doing:

$n$ choices to pick card $1$, $(n-1)$ possible places to put it

$n-1$ choices to pick card $2$, $(n-2)$ possibilities

$2$ choices to pick Card $n$, $1$ possibility to put it in the deck

edited Mar 09 '16 at 18:51

asked Mar 09 '16 at 18:48

GRS

4

You are asking about derangements. – N. F. Taussig Mar 09 '16 at 18:51
Thanks :) , I'll have a look – GRS Mar 09 '16 at 18:54
3

Notice that if you put card $1$ in position $2$, you now have $n-1$ places you can put card $2$. That makes the formula not so simple. But the link to "derangements" should help. – David K Mar 09 '16 at 18:55
1

See these two questions on recursive formulas for $d_n$. – JMoravitz Mar 09 '16 at 20:30
Thanks for you answers I maganed to solve it, the articles were helpful :) – GRS Mar 09 '16 at 20:39

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