nCr=n!(n−r)! r!{}^{n}C_{r} = \frac{n!}{(n-r)!\,r!}nCr=(n−r)!r!n!
nCr−1=n!(n−(r−1))! (r−1)!{}^{n}C_{r-1} = \frac{n!}{(n-(r-1))!\,(r-1)!}nCr−1=(n−(r−1))!(r−1)!n!
nCrnCr−1=n!(n−r)! r!÷n!(n−(r−1))! (r−1)!\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}} = \frac{n!}{(n-r)!\,r!} \div \frac{n!}{(n-(r-1))!\,(r-1)!}nCr−1nCr=(n−r)!r!n!÷(n−(r−1))!(r−1)!n!
= n!(n−r)! r!×(n−(r−1))! (r−1)!n!\frac{n!}{(n-r)!\,r!} \times \frac{(n-(r-1))!\,(r-1)!}{n!}(n−r)!r!n!×n!(n−(r−1))!(r−1)!
= (n+1−r)! (r−1)!(n−r)! r!\frac{(n+1-r)!\,(r-1)!}{(n-r)!\,r!}(n−r)!r!(n+1−r)!(r−1)!
= (n+1−r)(n−r)! (r−1)!(n−r)! r (r−1)!\frac{(n+1-r)(n-r)!\,(r-1)!}{(n-r)!\,r\,(r-1)!}(n−r)!r(r−1)!(n+1−r)(n−r)!(r−1)!
= n+1−rr\frac{n+1-r}{r}rn+1−r