**Value of (n 1)(n 2)....2n**

To find the value of the expression (n 1)(n 2)....2n, we need to understand the pattern and simplify it.

**Understanding the Pattern**

Let's consider a few values of n and observe the pattern:

For n = 1, the expression (n 1)(n 2)....2n = (1 1) = 1.

For n = 2, the expression (n 1)(n 2)....2n = (1 1)(2 2) = 1 * 2 = 2.

For n = 3, the expression (n 1)(n 2)....2n = (1 1)(2 2)(3 3) = 1 * 2 * 3 = 6.

For n = 4, the expression (n 1)(n 2)....2n = (1 1)(2 2)(3 3)(4 4) = 1 * 2 * 3 * 4 = 24.

We can observe that the value of the expression is the factorial of n. The factorial of a number n is the product of all positive integers less than or equal to n.

**Simplifying the Expression**

The expression (n 1)(n 2)....2n can be simplified as follows:

(n 1)(n 2)....2n = (1 1)(2 2)(3 3)....((n-1)(n-1))(n n)(n+1 n+1)....(2n 2n)

Now, we can pair each term in the expression:

(1 1)(2 2)(3 3)....((n-1)(n-1))(n n)(n+1 n+1)....(2n 2n) = [(1 2n)(2 2n-1)(3 2n-2)....((n-1) n+2)(n n+1)]

We can observe that each pair has the same product:

(1 2n) = 2n
(2 2n-1) = 2n-1
(3 2n-2) = 2n-2
...
((n-1) n+2) = n+2
(n n+1) = n+1

So, the simplified expression becomes:

[(1 2n)(2 2n-1)(3 2n-2)....((n-1) n+2)(n n+1)] = (2n)(2n-1)(2n-2)....(n+2)(n+1)

**Conclusion**

In conclusion, the value of the expression (n 1)(n 2)....2n is equal to the factorial of n, which can be simplified as (2n)(2n-1)(2n-2)....(n+2)(n+1). This expression represents the product of all positive integers from 1 to 2n.