Question

Sendo k = binomial de a sobre b; m= binomial de a sobre b+1;n = binomial de a sobre b+2. Utilize a relação de Stiffel e determine o valor do binomial de a+2 sobre b+2 utilizando as incógnitas acima.

Answer 1

$k=\left(\begin{array}{ccc}a\\b\\\end{array}\right) ; m= \left(\begin{array}{ccc}a\\b+1\\\end{array}\right);n= \left(\begin{array}{ccc}a\\b+2\\\end{array}\right) \\ \\ k+m= \left(\begin{array}{ccc}a\\b\\\end{array}\right)+ \left(\begin{array}{ccc}a\\b+1\\\end{array}\right)= \left(\begin{array}{ccc}a+1\\b+1\\\end{array}\right) \\ \\ m+n= \left(\begin{array}{ccc}a\\b+1\\\end{array}\right)+ \left(\begin{array}{ccc}a\\b+2\\\end{array}\right)=[tex] \left(\begin{array}{ccc}a+1\\b+2\\\end{array}\right) \\ \\ (k+m)+(m+n)= \left(\begin{array}{ccc}a+1\\b+1\\\end{array}\right)+ \left(\begin{array}{ccc}a+1\\b+2\\\end{array}\right)= \left(\begin{array}{ccc}a+2\\b+2\\\end{array}\right)$

Logo: $\left(\begin{array}{ccc}a+2\\b+2\\\end{array}\right)=2m+k+n$