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6.Permutation and Combination
normal
For $2 \le r \le n,\left( {\begin{array}{*{20}{c}}n\\r\end{array}} \right) + 2\,\left( \begin{array}{l}\,\,n\\r - 1\end{array} \right)$ $ + \left( {\begin{array}{*{20}{c}}n\\{r - 2}\end{array}} \right)$ is equal to
A
$\left( {\begin{array}{*{20}{c}}{n + 1}\\{r - 1}\end{array}} \right)$
B
$2\,\left( {\begin{array}{*{20}{c}}{n + 1}\\{r + 1}\end{array}} \right)$
C
$2\,\left( {\begin{array}{*{20}{c}}{n + 2}\\r\end{array}} \right)$
D
$\left( {\begin{array}{*{20}{c}}{n + 2}\\r\end{array}} \right)$
(IIT-2000)
Solution
(d) Expression $ = {\,^n}{C_r} + 2\,.{\,^n}{C_{r – 1}}{ + ^n}{C_{r – 2}}$
$ = {(^n}{C_r} + {\,^n}{C_{r – 1}}) + {(^n}{C_{r – 1}} + {\,^n}{C_{r – 2}})$
$ = {\,^{n + 1}}{C_r} + {\,^{n + 1}}{C_{r – 1}} = {\,^{n + 2}}{C_r}$.
Standard 11
Mathematics
