MCQ
$\sum\limits_{r = 0}^m {^{n + r}{C_n} = } $
- ✓$^{n + m + 1}{C_{n + 1}}$
- B$^{n + m + 2}{C_n}$
- C$^{n + m + 3}{C_{n - 1}}$
- DNone of these
we have $\sum\limits_{r = 0}^m {^{n + r}{C_n}} = \sum\limits_{r = 0}^m {^{n + r}{C_r}} { = ^n}{C_0}{ + ^{n + 1}}{C_1}{ + ^{n + 2}}{C_2} + ......{ + ^{n + m}}{C_m}$
$ = [1 + (n + 1)]{ + ^{n + 2}}{C_2}{ + ^{n + 3}}{C_3} + ........{ + ^{n + m}}{C_m}$
${ = ^{n + m + 1}}{C_{n + 1}}$ $[\because {\;^n}{C_r}{ = ^n}{C_{n - r}}]$
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