Mathematics › Binomial Theorem › Terms of Binomial Expansion
For $r=0,1, \ldots, 10$, let $A_r, B_r$ and $C_r$ denote, respectively, the coefficient of $x^r$ in the…
For $r=0,1, \ldots, 10$, let $A_r, B_r$ 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 $\sum_{r=1}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right)$ is equal to
$B_{10}-C_{10}$ $A_{10}\left(B_{10}^2-C_{10} A_{10}\right)$ 0 $C_{10}-B_{10}$ Solution
$A_r=$ Coefficient of $x^r$ in
$
\begin{aligned}
& (1+x)^{10}={ }^{10} C_r \\
& B_r=\text { Coefficient of } x^r \text { in } \\
& (1+x)^{20}={ }^{20} C_r \\
& C_r=\text { Coefficient of } x^r \text { in } \\
& (1+x))^{30}={ }^{30} C_r \\
& \therefore \sum_{r=1}^{10} A_r\left(B_{10} B_r-C_{10} A_r\right) \\
& =\sum_{r=1}^{10} A_r B_{10} B_r-\sum_{r=1}^{10} A_r C_{10} A_r \\
& =\sum_{r=1}^{10}{ }^{10} C_r{ }^{20} C_{10}{ }^{20} C_r \\
& -\sum_{r=1}^{10}{ }^{10} C_r{ }^{30} C_{10}{ }^{10} C_r \\
& =\sum_{r=1}^{10}{ }^{10} C_{10-r}{ }^{20} C_{10}{ }^{20} C_r \\
& -\sum_{r=1}^{10}{ }^{10} C_{10-r}{ }^{30} C_{10}{ }^{10} C_r \\
&
\end{aligned}
$
$
\begin{aligned}
& ={ }^{20} C_{10} \sum_{r=1}^{10}{ }^{10} C_{10-r} \cdot{ }^{20} C_r \\
& \quad-{ }^{30} C_{10} \sum_{r=1}^{10}{ }^{10} C_{10-r}{ }^{10} C_r \\
& ={ }^{20} C_{10}\left({ }^{30} C_{10}-1\right)-{ }^{30} C_{10}\left({ }^{20} C_{10}-1\right) \\
& ={ }^{30} C_{10}-{ }^{20} C_{10}=C_{10}-B_{10}
\end{aligned}
$
Asked in: JEE Advanced 2010 (Paper 2)
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