B.3 Expaniosn of \(^nC_r\)

\[\begin{eqnarray} ^{n+1}C_r &=& ^nC_r + ^nC_{r-1} \nonumber \\ \Rightarrow ^nC_r &=& ^{n-1}C_r + ^{n-1}C_{r-1} \nonumber \\ \Rightarrow ^{n+1}C_r &=& ^{n-1}C_r + ^{n-1}C_{r-1} + ^{n-1}C_{r-1} + ^{n-1}C_{r-2} \nonumber \\ \Rightarrow ^{n+1}C_r &=& ^{n-1}C_r + 2^{n-1}C_{r-1} + ^{n-1}C_{r-2} \nonumber \\ \end{eqnarray}\]

• Similarly, \(^nC_r = ^{n-2}C_r + 2^{n-2}C_{r-1} + ^{n-2}C_{r-2}\)