Just by definition of the factorial,

n!= (n-1)! \*n

If you want to find a common denominator of

a/n! + b/(n-1)!,

you can use this to see that n! is their common denominator and

a/n! + b/(n-1)! = a/((n-1)!\*n) + (b\*n)/((n-1)! \*n) = (a+bn)/ n!

In your reference, this trick happens twice, once with n! and (n-1)! and the other time with (m-n)! and (m-n-1)!.