On the Wikipedia page on derangements, the following description is given about how to count derangements:
Suppose that there are $n$ persons numbered $1,2,\ldots,n$. Let there be $n$ hats also numbered $1,2,\ldots,n$. We have to find the number of ways in which no one gets the hat having same number as his/her number. Let us assume that first person takes the hat $i$. There are $n-1$ ways for the first person to choose the number $i$. Now there are 2 options:
A. Person $1$ does not take the hat $i$. This case is equivalent to solving the problem with $n − 1$ persons $n − 1$ hats: each of the remaining $n − 1$ people has precisely 1 forbidden choice from among the remaining $n − 1$ hats ($i$'s forbidden choice is hat $1$).
B. Person $1$ takes the hat $i$. Now the problem reduces to $n − 2$ persons and $n − 2$ hats.
Aren't the two bolded statements in contradiction? Isn't the "first person" and "person 1" the same person? Is this explanation misworded?
