Permuting the digits of a 6 digit number with restrictions

Asked Jun 30 '22 at 18:39

Active Jun 30 '22 at 19:34

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How many ways are there to permute the digits of the number 123456 such that:

$n$ is not the $(n+1)$th digit for $1\leq n\leq 5$ and 6 is not the 1st digit.

I am not sure about how I should approach the problem without using some ugly casework. If there is an alternative method could someone give me a hint?

edited Jun 30 '22 at 19:34

Peter Phipps

3,065

asked Jun 30 '22 at 18:39

my_username

Are you familiar with the Inclusion-Exclusion Principle? What have you tried? – N. F. Taussig Jun 30 '22 at 19:07
4

You are looking for derangements of "612345" – Daniel Mathias Jun 30 '22 at 19:25
See this article for an introduction to Inclusion-Exclusion. Then, see this answer for an explanation of and justification for the Inclusion-Exclusion formula. Following the syntax of the 2nd link, I suggest letting $S_k$ denote the subset of numbers where $k$ is located in the $(k+1)$ digit place, thus violating the constraint. – user2661923 Jun 30 '22 at 19:58

Permuting the digits of a 6 digit number with restrictions

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