Let n be an integer greater than 1. How many distinct functions are there with domain (0..n] and codomain (0..2n] such that for any element x in the domain, the function value f(x) is strictly larger than x? Express the result in closed form.

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,

Explore the combinatorial problem of counting distinct functions with domain (0..n] and codomain (0..2n] where f(x) > x for each element x. This problem involves key concepts in combinatorics, including the use of inequalities and the Fundamental Principle of Counting. Suitable for undergraduate students.

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