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  4. The value of 30 C0- 30 C1+ 30 C2- 30 C3+ ldots+ 30 C10 is

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Q. The value of ${ }^{30} C_{0}-{ }^{30} C_{1}+{ }^{30} C_{2}-{ }^{30} C_{3}+\ldots+{ }^{30} C_{10}$ is

Binomial Theorem

Solution:

${ }^{30} C_{0} -{ }^{30} C_{1}+{ }^{30} C_{2}-{ }^{30} C_{3}+\ldots+{ }^{30} C_{10} $
= Coefficient of $ x^{10} $ in $(1-x)^{30}\left(1+x+x^{2}+\ldots+x^{10}+\ldots\right) $
= Coefficient of $ x^{10} $ in $(1-x)^{30}(1-x)^{-1} $
=Coefficient of $x^{10} $ in $(1-x)^{29} $
$={ }^{29} C_{10} $