What Is the Origin of the Last x! in the Factorial Equation (x + y)!?

[M. next]

Why is it that (x + y)!=(x + y)(x + y - 1)(x + y - 2)...(x + 1)x!
Where did the last "x!" come from?

Thanks

 

[jbunniii]

$$(x+y)! = (x+y)(x+y-1)(x+y-2)\ldots(x+1)(x)(x-1)\ldots(2)(1)$$
Now just rewrite the rightmost factors $(x)(x-1)\ldots(2)(1)$ as $x!$.

 

[M. next]

Thank you for your quick reply. I got the form you required, but still I have the concept missing. If you don't mind explaining why did we multiply by (x+1)(x)(x−1)…(2)(1)? It seems like we get to a place where y disappears by subtraction but then again why did we add the term (x+1) and so on?

 

[jbunniii]

Thank you for your quick reply. I got the form you required, but still I have the concept missing. If you don't mind explaining why did we multiply by (x+1)(x)(x−1)…(2)(1)? It seems like we get to a place where y disappears by subtraction but then again why did we add the term (x+1) and so on?

The definition of the factorial of any number $n$ is $(n)(n-1)\ldots(2)(1)$, i.e., you must keep subtracting until you get all the way down to $1$. Therefore, when calculating $(x+y)!$, you don't stop when you get to $x$; you must continue all the way to $1$.

 

[jbunniii]

Try it with a concrete example if it's still unclear. For example, if $x = 3$ and $y = 4$, then $x+y = 7$, and $(x+y)! = 7! = (7)(6)(5)(4)(3)(2)(1) = (7)(6)(5)(4)3!$.

 

[M. next]

Thank you a lot!