We are tasked with finding how many distinct functions $f$ there are with domain $\{1, 2, \dots, n\}$ and codomain $\{1, 2, \dots, 2n\}$, where for each element $x$ in the domain, the function value $f(x)$ satisfies $f(x) > x$. Let's break this down step by step:

Step 1: Define the domain and codomain

• The domain of the function is $\{1, 2, \dots, n\}$, which contains $n$ elements.
• The codomain of the function is $\{1, 2, \dots, 2n\}$, which contains $2n$ elements.

Step 2: Restriction on the function values

For each $x \in \{1, 2, \dots, n\}$, the function must satisfy: $f(x) > x$ This means that $f(x)$ cannot take values in $\{1, 2, \dots, x\}$. Instead, $f(x)$ must be selected from $\{x+1, x+2, \dots, 2n\}$.

Step 3: Count the available choices for each $x$

For each $x$, the available values for $f(x)$ are: $\{x+1, x+2, \dots, 2n\}$ The number of elements in this set is $2n - x$, because the largest value is $2n$ and the smallest value is $x+1$. Thus, for each $x \in \{1, 2, \dots, n\}$, there are $2n - x$ choices for $f(x)$.

Step 4: Total number of functions

To find the total number of distinct functions, we multiply the number of choices for each $f(x)$ for $x = 1, 2, \dots, n$. Therefore,