We know that if $^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}$ are binomial coefficients then
$(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3} + \dots + C_{n} x^{n}$ . Various relations among binomial coefficients can be derived by putting $x = 1, - 1, x = i, x = w, where, i = \sqrt{- 1}, w = - \frac{1}{2} + \frac{i \sqrt{3}}{2}$ Some other identities can be derived by adding and subtracting two such identities. The expression $(a + i b)^{n}$ can be evaluated by using De-Moiver's theorem by putting $a = r cos θ, b = r sin θ$ .

The value of $^{n} C_{0} -^{n} C_{2} +^{n} C_{4} -^{n} C_{6} + \dots$ must be

2^{n / 2} cos \frac{n π}{2}

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2^{n / 2} sin \frac{n π}{2}

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2^{n / 2} cos \frac{n π}{4}

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2^{n / 2} sin \frac{n π}{4}

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Solution

The correct option is C $2^{n / 2} cos \frac{n π}{4}$

$(1 + i)^{n} = C_{0} + C_{1} i + C_{2} i^{2} + C_{3} i^{3} + . . . + C_{n} i^{n}$
$= (C_{0} - C_{2} + C_{4} - C_{6} + . . . + i (C_{1} - C_{3} + C_{5} - . . .$

$\Rightarrow C_{0} - C_{2} + C_{4} - C_{6} + . . .$ is the real part of $(1 + i)^{n}$

= Real part of ${(\sqrt{2} (cos \frac{π}{4} + i sin \frac{π}{4}))}^{n}$

= Real part of $2^{n / 2} (cos \frac{n π}{4} + i sin \frac{n π}{4})$

$= 2^{n / 2} cos \frac{n π}{4}$

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Similar questions

^{n} C_{0},^{n} C_{1},^{n} C_{2}, \dots,^{n} C_{n}

(1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + C_{3} x^{3} + \dots + C_{n} x^{n}

x = 1, - 1, x = i, x = w, where, i = \sqrt{- 1}, w = - \frac{1}{2} + \frac{i \sqrt{3}}{2}

(a + i b)^{n}

a = r cos θ, b = r sin θ

(^{n} C_{0} -^{n} C_{2} +^{n} C_{4} -^{n} C_{6} + \dots)^{2} + (^{n} C_{1} -^{n} C_{3} +^{n} C_{5} \dots)^{2}

The value of $^{n} C_{0}$ - $^{n} C_{2}$ + $^{n} C_{4}$ - $^{n} C_{6}$ . . . . .

Which of the following is/are true?

(1) $(1 + i)^{n}$ = $2^{\frac{n}{2}}$ $(s i n \frac{n π}{4} + i c o s \frac{n π}{4})$

If $(1 + i)^{n}$ = $^{n} C_{0}$ + $^{n} C_{1}$ i - $^{n} C_{2}$ - $^{n} C_{3}$ i + $^{n} C_{4}$ ...............

(2) $^{n} C_{0}$ - $^{n} C_{2}$ + $^{n} C_{4}$ - $^{n} C_{6}$ ................ = $2^{\frac{n}{2}}$ $s i n \frac{n π}{4}$

(3) $^{n} C_{1}$ - $^{n} C_{3}$ + $^{n} C_{5}$ - $^{n} C_{7}$ ....................= $2^{\frac{n}{2}} c o s \frac{n π}{4}$

^{2 n + 1} C_{0} +^{2 n + 1} C_{1} +^{2 n + 1} C_{2} + \dots +^{2 n + 1} C_{n} = 4^{11}

1 + x)^{n} = C_{0} + C_{1} x + C_{2} x^{2} + . . . . . . . + C_{n} x^{n}

C_{0} - C_{2} + C_{4} - C_{6} + . . . . . . .

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