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Chapter 9: Problem 32

Simplify the ratio of factorials. $$ \frac{25 !}{23 !} $$

Short Answer

Expert verified
The simplified ratio is \(25*24\), or \(600\).

Step by step solution

01

Breaking Down Larger Factorial

Express the larger factorial (25!) in terms of the smaller factorial (23!). Using the idea that the factorial of any number n is equivalent to n*(n-1)!, we can express 25! as 25*24*23!.
02

Simplify the Ratio

Simplify the ratio by cancelling common factors. In this case, you'll be able to cancel out 23! from the numerator (top) and the denominator (bottom). This will leave you with 25*24.
03

Calculate Remaining Product

Calculate the remaining product left after cancellation, which is 25*24.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ratio of Factorials
In mathematics, the ratio of factorials represents a division of one factorial by another. Generally, a factorial is the product of all positive integers from 1 to a given number \( n \), denoted as \( n! \). For instance, calculating the ratio \( \frac{25!}{23!} \) involves dividing 25 factorial by 23 factorial.
  • Factorials grow very fast, making direct computation cumbersome and lengthy.
  • This rapid growth is why expressing larger factorials in terms of a smaller one can significantly ease the calculation.
  • In our example, instead of computing 25! and 23! directly and then dividing, we simplify by breaking down the larger factorial (25!).
Knowing the ratio of factorials helps in simplifying expressions, particularly in combinatorics, where such scenarios frequently occur.
Simplification of Factorial Expressions
Simplifying factorial expressions is crucial to make calculations manageable. It involves canceling equivalent terms in the numerator and denominator of the expression. Let's explore how to approach the simplification.
  • Given the expression \( \frac{25!}{23!} \), you express 25! in terms of 23! as 25 \( \times \) 24 \( \times \) 23!.
  • By doing this, you can cancel the common term \( 23! \), leaving you with a much simpler expression: \( 25 \times 24 \).
  • This step avoids unnecessary multiplication of all integers involved, smartly reducing the problem size.
Thus, simplifying factorial expressions not only saves time but also minimizes errors in calculations especially when dealing with large numbers.
Multiplicative Properties of Factorials
One of the concepts that make factorial operations interesting is the way they utilize multiplication properties. Each factorial is essentially a series of multiplicative steps.
  • Factorials leverage the property that each successive product builds on the last, meaning \( n! = n \times (n-1)! \).
  • This property is particularly useful for simplifying when comparing or dividing different factorial values.
  • In a scenario like \( \frac{25!}{23!} \), this principle of building on previous products facilitates the process of simplification.
Employing the multiplicative nature of factorials allows us to simplify complex expressions effectively, proving vital in scenarios like permutations and combinations.

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Most popular questions from this chapter

True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Consider the sequence \(\sqrt{6}, \sqrt{6+\sqrt{6}}, \sqrt{6+\sqrt{6+\sqrt{6}}}, \ldots\) (a) Compute the first five terms of this sequence. (b) Write a recursion formula for \(a_{n}\), for \(n \geq 2\). (c) Find lim \(a_{n}\).

Consider the sequence \(\left\\{a_{n}\right\\}=\left\\{n r^{n}\right\\}\). Decide whether \(\left\\{a_{n}\right\\}\) converges for each value of \(r\). (a) \(r=\frac{1}{2}\) (b) \(r=1\) (c) \(r=\frac{3}{2}\) (d) For what values or \(r\) does the sequence \(\left\\{n r^{n}\right\\}\) converge?

Find two divergent series \(\sum a_{n}\) and \(\sum b_{n}\) such that \(\Sigma\left(a_{n}+b_{n}\right)\) converges.

State the guidelines for finding a Taylor series.

Suppose that \(\sum a_{n}\) diverges and \(c\) is a nonzero constant. Prove that \(\Sigma c a_{n}\) diverges.
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