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  4. The expressions ( 10 C 0)2-( 10 C 1)2+( 10 C 2)2- ldots ldots .+( 10 C 8)2-( 10 C 9)2+( 10 C 10)2 equals

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Q. The expressions $\left({ }^{10} C _0\right)^2-\left({ }^{10} C _1\right)^2+\left({ }^{10} C _2\right)^2-\ldots \ldots .+\left({ }^{10} C _8\right)^2-\left({ }^{10} C _9\right)^2+\left({ }^{10} C _{10}\right)^2$ equals

Binomial Theorem

Solution:

We have
$(1+ x )^{10}={ }^{10} C _0+{ }^{10} C _1 x +{ }^{10} C _2 x ^2+{ }^{10} C _3 x ^3+\ldots \ldots . .+{ }^{10} C _9 x ^9+{ }^{10} C _{10} x ^{10} $....(1)
$\text { Also }( x -1)^{10}={ }^{10} C _0 x ^{10}-{ }^{10} C _1 x ^9+{ }^{10} C _2 x ^8+\ldots \ldots . .{ }^{10} C _9 x +{ }^{10} C _{10} $....(2)
$\text { Multiplying (1) and }(2) \text {, we get } $
$\left( x ^2-1\right)^{10}=\left({ }^{10} C _0+{ }^{10} C _1 x +\ldots \ldots .+{ }^{10} C _9 x ^9+{ }^{10} C _{10} x ^{10}\right)$
$\left({ }^{10} C _0 x ^{10}+{ }^{10} C _1 x ^9+\ldots \ldots .-{ }^{10} C _9 x +{ }^{10} C _{10}\right)$
Comparing the coefficients of $x^{10}$ in (3), we get
${ }^{10} C _5(-1)^5=\left({ }^{10} C _0\right)^2-\left({ }^{10} C _1\right)^2+\left({ }^{10} C _2\right)^2-\ldots \ldots .-\left({ }^{10} C _9\right)^2+\left({ }^{10} C _{10}\right)^2$