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  4. 10 C0 20 C10- 10 C1 18 C10+ 10 C2 16 C10-⋅s=

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Q. ${ }^{10} C_{0}{ }^{20} C_{10}-{ }^{10} C_{1}{ }^{18} C_{10}+{ }^{10} C_{2}{ }^{16} C_{10}-\cdots=$

Binomial Theorem

Solution:

${ }^{10} C_{0}{ }^{20} C_{10}-{ }^{10} C_{1}{ }^{18} C_{10}+{ }^{10} C_{2}{ }^{16} C_{10}-\cdots$
$=$ coefficient of $x^{10}$ in $\left[{ }^{10} C_{0}(1+x)^{20}-{ }^{10} C_{1}(1+x)^{18}\right.
\left.+{ }^{10} C_{2}(1+x)^{16}-\cdots\right]$
$=$ coefficient of $x^{10}$ in $\left[{ }^{10} C_{0}\left((1+x)^{2}\right)^{10}-{ }^{10} C_{1}\left((1+x)^{2}\right)^{9}+\right. \left.{ }^{10} C_{2}\left((1+x)^{2}\right)^{8}-\cdots\right]$
$=$ coefficient of $x^{10}$ in $\left[(1+x)^{2}-1\right]^{10}$
$=$ coefficient of $x^{10}$ in $\left[2 x+x^{2}\right]^{10}$
$=2^{10}$