In the attached document, page 7, for the second order incremental ADC, factor (n-i+1) is derived and used to multiply with comparator output in updown counter in order to implement 2nd order digital filter

Now i want to increase order of digital filter and i tried to derive the factor for 3rd order digital filter implementation and got  (n+2-i)*(n+1-i)/2

can someone verify this factor?