Question

For $r = 0, 1, ... , 10,$ if $A_r,B_r$ and $C_r$ denote respectively the coefficient of $x^r$ in the expansions of $(1 + x)^{10}, (1 + x)^{20}$ and $(1 + x)^{30}$. Then, $\displaystyle\sum_{r = 0}^{10} A_r(B_{10}B_r-C_{10}A_r)$ is equal to

• A. $B_{10} -C_{10}$
• B. $A_{10}(B^2_{10}-C_{10}A_{10})$
• C. $0$
• D. $C_{10}-B_{10}$

Answer 1

Answer: (D)

$A_r$ = Coefficient of $x^r$ in $(1 + x)^{10} = ^{10}C_r$
$B_r$= Coefficient of $x^r$ in $(1+x)^{20} = ^{20}C_r$
$C_r$= Coefficient of $x^r$ in $(1+x)^{30} =^{30}C_r$
$\therefore \displaystyle\sum_{r=1}^{10} A_r (B_{10}B_r-C_{10}A_r) =\displaystyle\sum_{r=1}^{10} A_r B_{10} B_r- \displaystyle\sum_{r=1}^{10} A_r C_{10}A_r $
=$\displaystyle \sum_{r=1}^{10} \,{}^{10}C_r \,{}^{20}C_{10} \,{}^{20}C_r - \displaystyle \sum_{r=1}^{10} \,{}^{10}C_r \,{}^{30}C_{10} \,{}^{10}C_r$
=$\displaystyle \sum_{r=1}^{10} \,{}^{10}C_{10-r} \,{}^{20}C_{10} \,{}^{20}C_r - \displaystyle \sum_{r=1}^{10} \,{}^{10}C_{10-r} \,{}^{30}C_{10} \,{}^{10}C_r$
=${}^{20}C_{10} \displaystyle \sum_{r=1}^{10} \,{}^{10} C_{10-r} . ^{20}C_r -\,{}^{30}C_{10} \displaystyle \sum_{r=1}^{10} \,{}^{10} C_{10-r} \,{}^{10}C_r$
$={}^{20}C_{10} (^{30}C_{10}-1)- \,{}^{30}C_{10}(^{20}C_{10}-1)$
$= {}^{30}C_{10} - {}^{20}C_{10} = C_{10}-B_{10}$