How was the common denominator arrived at in this example?

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)!.
The first denominator is equal to n (n-1)! (m-n)!
The second denominator is equal to (m-n+1) (m-n)! (n-1)!
The common denominator is equal to n (n-1)! (m-n+1) (m-n)!
by

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