# Solving an Unknown Factorial

• SprucerMoose
In summary: However, you can use a calculators stirling approximation to approximate the value of n! In summary, there is no elegant method to solve an unknown factorial algebraically, but you can use a calculators stirling approximation to approximate the value of n!

#### SprucerMoose

Is there a method to solve an unknown factorial algebraically?

## Homework Statement

5040 = a!

I found the solution a = 7 through trial and error, was just wondering if a more elegant method exists.

what you can do for big n is to apply stirling approximation
ln(n!)=(n*ln(n)) - n
so you can make a handy table or chart to pluck up the corresponding n values, apart from that doing this is a real mess.

Thanks a lot

Or, if you want an "algorithm" rather than "trial and error",

5040/2= 2520

2520/3= 840

840/4= 210

210/5= 42

42/6= 7
and

7/7= 1.

Hi, question to @HallsofIvy, what is the procedure if the number is not wholly divisible at some point. For example you take the number 5 and try to divide it by 2, you get 2 remainder 1. How would you proceed from here? I vaguely remember doing something with the remainder, but don't remember what it was. Searching on google has not produced the results I'm seeking. Please let me know if you know of a good resource for this or if you know how to proceed. Thank you in advance!

silverstar said:
Hi, question to @HallsofIvy, what is the procedure if the number is not wholly divisible at some point. For example you take the number 5 and try to divide it by 2, you get 2 remainder 1. How would you proceed from here? I vaguely remember doing something with the remainder, but don't remember what it was. Searching on google has not produced the results I'm seeking. Please let me know if you know of a good resource for this or if you know how to proceed. Thank you in advance!
Do you know the definition of " n! " ?

SammyS's point is that n! is, by definition, divisible by all those numbers.

abhishek ghos said:
what you can do for big n is to apply stirling approximation
ln(n!)=(n*ln(n)) - n
so you can make a handy table or chart to pluck up the corresponding n values, apart from that doing this is a real mess.

Don't forget the constant: n! ~ sqrt(2*pi)*n^(n + 1/2) * exp(-n). If you don't have the sqrt(2*pi) factor the results are inaccurate. Here are the results for log(n!). Column 1 is n, column 2 is log(n!), column 3 is the log of approximation with the sqrt(2pi) factor, and column 4 is the log without the factor:
1 0.0000 -0.0811 -1.0000
2 0.6931 0.6518 -0.2671
3 1.7918 1.7641 0.8451
4 3.1781 3.1573 2.2383
5 4.7875 4.7708 3.8519
6 6.5793 6.5654 5.6464
7 8.5252 8.5133 7.5943
8 10.6046 10.5942 9.6753
9 12.8018 12.7926 11.8736
10 15.1044 15.0961 14.1771

As to how to solve your equation x! = y if you are not sure whether or not there is an integer n giving n! = y: write an equation in the Gamma function, so that x! is defined for all x > 0 (whether integer or not). If you have good software available, your equation can be dealt with using standard techniques such as Newton's method, etc.

RGV

## What is an unknown factorial?

An unknown factorial is a mathematical problem where you are given the value of a factorial expression and you have to find the value of the unknown number.

## How do I solve an unknown factorial?

To solve an unknown factorial, you can use various methods such as using the inverse of the factorial function, using algebraic manipulation, or using a calculator or computer program.

## What are some tips for solving an unknown factorial?

Some tips for solving an unknown factorial include trying different methods, breaking the problem into smaller parts, and checking your solution to make sure it satisfies the given factorial expression.

## Can an unknown factorial have more than one solution?

Yes, an unknown factorial can have more than one solution. This can happen when the given factorial expression has multiple factors or when there are multiple ways to simplify the expression.

## What real-life applications use unknown factorials?

Unknown factorials are used in various fields such as physics, engineering, and computer science to solve problems involving combinations and permutations, as well as in probability and statistics to calculate probabilities and combinations. They are also used in cryptography and coding theory to generate secure codes and in machine learning algorithms.