Q. 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, $\displaystyle \sum A_r(B_{10}B_r-C_{10}A_r)$ is equal to

Solution:

$A_r$ = 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 $
=$\displaystyle \sum_{r=1}^{10} \, ^{10}C_r \, ^{20}C_{10} \, ^{20}C_r - \displaystyle \sum_{r=1}^{10} \, ^{10}C_r \, ^{30}C_{10} \, ^{10}C_r$
=$\displaystyle \sum_{r=1}^{10} \, ^{10}C{10_r} \, ^{20}C_{10} \, ^{20}C_r - \displaystyle \sum_{r=1}^{10} \, ^{10}C{10_r} \, ^{30}C_{10} \, ^{10}C_r$
=$^{20}C_{10} \displaystyle \sum_{r=1}^{10} \, ^{10} C_{10-r} . ^{20}C_r -\, ^{30}C_{10} \displaystyle \sum_{r=1}^{10} \, ^{10} C_{10-r} \, ^{10}C_r$
$=^{20}C_{10} (^{30}C_{10}-1)- \, ^{30}C_{10}(^{20}C_{10}-1)$
$= ^{30}C_{10} - ^{20}C_{10} = C_{10}-B_{10}$

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