Question

The coefficient of $x^m$ in $(1+x{)}^k+(1+x{)}^{k+1}+..............+(1+x{)}^n,$$(k\le m\le n)$, is given by:

• A. ⁿ⁺¹C_m
• B. ⁿC_m
• C. ^n-1C_m-1
• D. ⁿ⁺¹C_m+1

Answer 1

The correct option is D

ⁿ⁺¹C_m+1

$(1+x{)}^k+(1+x{)}^{k+1}+..............+(1+x{)}^n,$

= $\frac{(1+x{)}^k\left(\right(1+x{)}^{n-k+1}-1)}{(1+x)-1}$

= $\frac{1}{x}(1+x{)}^{n+1}-(1+x{)}^k$.

$\therefore$ Coeff. of $x^m$ = Coeff. of $x^{m+1}$ in

$(1+x{)}^{n+1}-(1+x{)}^k$.

= ${}^{n+1}{C}_{m+1}(m+1>k)$.