How do you simplify the factorial expression $\frac{(n + 2)!}{n!}$ $((n+2)!)/(n!)$ ?

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Answer 1 · 2017-02-08T00:22:56

$\frac{(n + 2)!}{n!} = (n + 2) (n + 1)$ $((n+2)!)/(n!) = (n+2)(n+1)$

Remember that:

$n! =n(n-1)(n-2)...1$

And so

$(n+2)! =(n+2)(n+1)(n)(n-1) ... 1$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \=(n+2)(n+1)n!$

So we can write:

$((n+2)!)/(n!) = ((n+2)(n+1)n!)/(n!)$
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =(n+2)(n+1)$

How do you simplify the factorial expression ((n+2)!)/(n!)(n+2)!n!((n+2)!)/(n!)?