[_pmatrix i,j = 0 i j pmatrix ^n ^nC_i,^nC_j ] is equal to

Question

$$\sum\limits_{\begin{pmatrix} {i,j = 0}  \
   {i \ne j}  \ \end{pmatrix} }^n {{}^n{C_i}\,{}^n{C_j}} $$ is equal to

Accepted Answer

$$\sum\limits_{i,\,j = 0\,\,i \ne j}^n {{}^n{C_i}\,{}^n{C_j} = \sum\limits_{i,\,j = 0}^n {{}^n{C_i}\,{}^n{C_j} - \sum\limits_{i = j}^n {{}^n{C_i}\,{}^n{C_j}} } } $$

$$ = \sum\limits_{j = 0}^n {{}^n{C_i}\,\sum\limits_{j = 0}^n {{}^n{C_j} - \sum\limits_{i = 0}^n {{}^n{C_i}\,{C_i}} } } $$

$$ = {2^n}\,.\,{2^n} - {}^{2n}{C_n}$$

$= {2^{2n}} - {}^{2n}{C_n}$

Solutions · 1