${^{10} C}_{o} + {^{10} C}_{2} + {^{10} C}_{4} + . . . . . + {^{10} C}_{10} =$

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Solution

We've,
$(1 + x)^{10} =^{10} C_{0} +^{10} C_{1} x + . . . . . . +^{10} C_{10} x^{10}$ ......(1).
Putting $x = 1$ in (1) we get,
$2^{10} =$ $^{10} C_{0} +^{10} C_{1} + . . . . . . +^{10} C_{10}$ ......(2).
Again putting $x = - 1$ in (1) we get,
$0 =$ $^{10} C_{0} -^{10} C_{1} + . . . . . . +^{10} C_{10}$ ......(3).
Now adding (2) and (3) we get,
$2^{10} =$ $2 (^{10} C_{0} +^{10} C_{2} + . . . . . . +^{10} C_{10})$
or, $(^{10} C_{0} +^{10} C_{2} + . . . . . . +^{10} C_{10}) = 2^{9}$ .

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