Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. n!n! = (2n)! for all positive integers n.

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Recall the definition of the factorial function: for a positive integer $n$, $n! = 1 \times 2 \times 3 \times \cdots \times n$.

Understand the expressions given: $n!n!$ means multiplying $n!$ by itself, while $(2n)!$ means the factorial of \$2n$, which is $1 \times 2 \times 3 \times \cdots \times (2n)$.

To check if $n!n! = (2n)!$ for all positive integers $n$, consider testing small values of $n$ to see if the equality holds.

For example, when $n=1$, $n!n! = 1! \times 1! = 1 \times 1 = 1$ and $(2n)! = (2 \times 1)! = 2! = 2$, so the equality does not hold here.

Since the equality fails for $n=1$, the statement is false for all positive integers $n$. This counterexample disproves the statement.

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Factorials and Their Properties

A factorial, denoted n!, is the product of all positive integers from 1 to n. Understanding how factorials grow and their basic properties is essential to compare expressions like n!·n! and (2n)!.

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Inequalities and Growth Rates of Factorials

Factorials grow very rapidly, and (2n)! grows faster than n!·n!. Recognizing this difference helps determine whether n!·n! equals (2n)! or not by comparing their magnitudes.

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Counterexamples in Mathematical Proof

To disprove a statement, providing a single counterexample suffices. Testing the equality for small values of n can quickly show whether the statement holds universally or not.

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