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  4. The coefficient of x5 in the expansion of (1+x)21 +(1+x)22+ ldots+(1+x)30 is

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Q. The coefficient of $x^{5}$ in the expansion of $(1+x)^{21} +(1+x)^{22}+\ldots+(1+x)^{30}$ is

AP EAMCETAP EAMCET 2017

Solution:

We know that, coefficient of $x^{r}$ in the expansion of $(1+x)^{n}$ is given by ${ }^{n} C_{r}$.
$\because$ Coefficient of $x^{5}$ in the expansion of
$(1+x)^{21}+(1+x)^{22}+\ldots+(1+x)^{2 n}$
$={ }^{21} C_{5}+{ }^{22} C_{5}+\ldots+{ }^{30} C_{5}$
$=\left({ }^{21} C_{6}+{ }^{21} C_{5}+{ }^{22} C_{5}+\ldots+{ }^{30} C_{5}\right)-{ }^{21} C_{6}$
$-\left({ }^{22} C_{6}+{ }^{22} C_{b}+\ldots+{ }^{30} C_{b}\right)-{ }^{21} C_{6}$
$\left[\because{ }^{n} C_{Y}+{ }^{n} C_{t-1}={ }^{n+1} C_{I}\right]$
$= \left.{ }^{23} C_{6}+{ }^{23} C_{5}+\ldots+{ }^{30} C_{6}\right)-{ }^{21} C_{6}$
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$=\left({ }^{30} C_{6}+{ }^{30} C_{5}\right)-{ }^{21} C_{6}$
$={ }^{31} C_{6}-{ }^{21} C_{6}$