Question

${}^n{C}_n{+}^{n+1}{C}_n{+}^{n+2}{C}_n+...{+}^{n+k}{C}_{n+1}=$

• A. ${}^{n+k-1}{C}_{n+1}$
• B. ${}^{n+k}{C}_{n+1}$
• C. ${}^{n+k+1}{C}_{n+1}$
• D. none of these

Answer 1

The correct option is C ${}^{n+k+1}{C}_{n+1}$

${}^n{C}_n$ can be neither as

${}^{n+1}{C}_{n+1}$

and now

${}^n{C}_n{+}^{n+1}{C}_n{+}^{n+2}{C}_n+...{+}^{n+k}{C}_{n+1}$

can be evaluated with

the property:

${}^n{C}_r{+}^n{C}_{r-1}{=}^{n+1}{C}_r$

${\therefore }^{n+1}{C}_{n+1}{+}^{n+1}{C}_n$

${=}^{n+2}{C}_{n+1}$

${\therefore }^{n+2}{C}_{n+1}{+}^{n+2}{C}_n+...{.}^{n+k}{C}_n$

Again : ${}^{n+2}{C}_{n+1}{+}^{n+2}{C}_{n+1}$

${=}^{n+3}{C}_{n+1}$

$\therefore$ This way

${}^{n+k}{C}_{n+1}{+}^{n+k}{C}_n$

${=}^{n+k+1}{C}_{n+1}$