The value of $^{4 n} C_{0} +^{4 n} C_{4} +^{4 n} C_{8} + . . . +^{4 n} C_{4 n}$ is

2^{4 n - 2} + (- 1)^{n} 2^{2 n - 1}

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2^{4 n - 2} + 2^{2 n - 1}

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2^{2 n - 2} + (- 1)^{n} 2^{4 n - 1}

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Solution

The correct option is A $2^{4 n - 2} + (- 1)^{n} 2^{2 n - 1}$
$(1 + 1)^{4 n} =^{4 n} C_{0} +^{4 n} C_{1} +^{4 n} C_{2} . . . . . . . . . . . .^{4 n} C_{4 n}$
$(1 - 1)^{4 n} =^{4 n} C_{0} -^{4 n} C_{1} +^{4 n} C_{2} . . . . . . . . . . .^{4 n} C_{4 n}$
$\Rightarrow^{4 n} C_{0} +^{4 n} C_{2} +^{4 n} C_{4} +^{4 n} C_{6} . . . . . . . . .^{4 n} C_{4 n}$
$2^{4 n - 1} (3)$
$(1 + i)^{4 n} =^{4 n} C_{0} +^{4 n} C_{1} +^{4 n} C_{2} i^{2} +^{4 n} C_{3} (i)^{3} + . . . . . . . . . . . . .^{4 n} C_{4 n} i^{4 n} (4)$
$^{4 n} C_{0} +^{4 n} C_{4} +^{4 n} C_{8} . . . . . . . . .^{4 n} C_{4 n} =$
$= 2^{4 n - 2} + 2^{2 n - 1} (- 1)^{n}$
$= 2^{(4 n - 2)} + 2^{(2 n - 1)} (- 1)^{n}$

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