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


b. (2n)! / (2n − 1)! = 2n

Verified step by step guidance

1. Recall the definition of factorial: for any positive integer \(k\), \(k! = k \times (k-1) \times (k-2) \times \cdots \times 1\).
2. Write out the factorial expressions explicitly: \((2n)! = (2n) \times (2n - 1) \times (2n - 2) \times \cdots \times 1\) and \((2n - 1)! = (2n - 1) \times (2n - 2) \times \cdots \times 1\).
3. Set up the fraction and simplify by canceling common terms: \(\frac{(2n)!}{(2n - 1)!} = \frac{(2n) \times (2n - 1) \times \cdots \times 1}{(2n - 1) \times \cdots \times 1} = 2n\).
4. Conclude that the statement \(\frac{(2n)!}{(2n - 1)!} = 2n\) is true because the factorial terms cancel except for the factor \$2n$ in the numerator.
5. Therefore, the expression simplifies exactly to \$2n$, confirming the statement is correct.

Key Concepts

Factorials and Their Properties

A factorial, denoted n!, is the product of all positive integers from 1 up to n. Understanding how factorials expand and simplify is essential, especially when dealing with ratios like (2n)! / (2n − 1)!, where many terms cancel out.

Simplifying Factorial Expressions

When dividing factorials such as (2n)! by (2n − 1)!, most terms cancel, leaving only the highest term, 2n. This simplification helps determine if the given expression equals 2n or not.

Verification Using Counterexamples

To test the truth of a statement involving factorials, substituting specific values of n can confirm or refute it. A single counterexample disproves the statement, while consistent results support its validity.