Fascinating Factorials: Exploring the Mystery of the Last x! in (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!