Exploring Solutions to a!b! = a! + b! + c^2 for Positive Integers a, b, and c

In summary, the conversation is about solving the equation a!b! = a! + b! + c^2 for positive integers a, b, and c. The participants discuss possible methods, with one suggesting brute force and the others considering alternative approaches. One participant suggests that c can be any positive integer that satisfies c = √(a!b! - a! - b!). Another participant suggests that a! can be rewritten as 1 + (a! + c^2)/b!, but does not have a clear solution.
  • #1
msudidi
2
0
Given that a, b, and c are positive integers solve the following equation.

a!b! = a! + b! + c^2

anyone?
 
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  • #2
I found the answer through brute force: a=2, b=3, c=2.

Not sure if there is a more elegant solution though.
 
  • #3
vorde, thanks for trying, I am getting the same answer too:smile:

but I'm seeking for method to solve it: how to relate multiplication of 2 factorials and their sums? if we can, then c wouldn't be a problem.

there should be a way to solve it:rolleyes:
 
  • #4
"Brute force" is a method! Please clarify what you are looking for.
 
  • #5
msudidi said:
Given that a, b, and c are positive integers solve the following equation.

a!b! = a! + b! + c^2

anyone?

Doesn't it work for all positive integers c such that [itex]c = \sqrt{a!b! -a! -b!}[/itex]? :biggrin:

Spit-balling here, we have [itex]a!b! = a! + b! + c^2[/itex]? Doesn't that imply that [itex]\displaystyle a! = 1 + \frac{a! + c^2}{b!} = 1 + \frac{a(a-1)(a-2)(a-3)...}{b(b-1)(b-2)...} + \frac{c^2}{b!} = 1 + \prod_{k = (b+1)}^a k + \frac{c^2}{b!}[/itex]. Don't know where I'm going with that...
 

What is the meaning of "Multiplication of Factorials"?

The multiplication of factorials is a mathematical operation that involves multiplying two or more numbers that are expressed as factorials. Factorials are denoted by an exclamation mark (!) and represent the product of all positive integers less than or equal to a given number.

How is the multiplication of factorials calculated?

The multiplication of factorials is calculated by first expanding the factorials into their numerical form and then multiplying the numbers together. For example, (4!) x (3!) = (4 x 3 x 2 x 1) x (3 x 2 x 1) = 24 x 6 = 144.

What is the difference between the multiplication of factorials and the multiplication of regular numbers?

The multiplication of factorials differs from the multiplication of regular numbers in that factorials involve multiplying all the numbers from 1 up to a given number, while regular multiplication only involves multiplying the given numbers. For example, 5! x 4! = 5 x 4 x 3 x 2 x 1 x 4 x 3 x 2 x 1 = 2880, while 5 x 4 = 20.

Can the multiplication of factorials be simplified?

Yes, the multiplication of factorials can be simplified by using the factorial property (n!) x (n+1) = (n+1)!. This property can be used to reduce the number of terms in the multiplication of factorials and make the calculation easier.

What are some real-life applications of the multiplication of factorials?

The multiplication of factorials is commonly used in probability and statistics to calculate the number of possible permutations or combinations of a set of objects. It is also used in the binomial theorem, which is used in binomial expansions and in the calculation of probabilities in genetics. In computer science, the multiplication of factorials is used in algorithms for efficiently calculating large factorials.

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