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In how many permutations of 1,..,n does the integer 1 precede the integer 2? (For instance, (1,3,2), as opposed to (2,3,1)).

I am trying to check my answer to this competition practice problem: (n-1)!

computerScientistD
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    And what is your answer? – lulu Dec 06 '23 at 19:02
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    As a general rule here, include your thoughts and attempts on problems in your initial post. Don't hold them back and wait for someone to have to ask for them. Doing so does many things, including but not limited to showing that you care, clarifying the problem and what tools/techniques you are able to use, highlighting what specific errors or misunderstandings you may have had so that responses can be better written to target your specific needs, and even in the case of truly difficult problems at least giving others a starting point to work from to make getting an answer easier. – JMoravitz Dec 06 '23 at 19:07
  • With regards to your attempted answer, no this is not correct. $(n-1)!$ is the number of permutations where $1$ immediately precedes the integer $2$, that there is no gap between $1$ and $2$, for example in $(1,2,3,4)$ or $(3,1,2,4)$ etc... but the examples here make it clear that we don't care about $1$ immediately preceding $2$, just that it occurs somewhere before $2$ possibly several spaces before $2$ like $(1,4,3,2)$ – JMoravitz Dec 06 '23 at 19:13
  • For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – Henrik supports the community Dec 06 '23 at 22:21

1 Answers1

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You can think of this way: the number $n_1$ of cases that 1 precedes 2 is equal of the number $n_2$ of cases that 2 precedes 1, cause there's nothing especial between these two numbers. But, these two situations are complementary, so:

$$ \left\{ \begin{array}{c} n_1 + n_2 = T \\ n_1 = n_2 \\ \end{array} \right. $$

With $T$ being the total number of cases, $T=n!$

Finally: $n_1=n_2=\frac{n!}{2}$

meu estudo
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