C₀² - C₁² + C₂² ... (-1) Cₙ² , where n is an even integer is

Correct Answer is : None of these

Correct Answer is : None of these

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JEE Mains Model Paper 15

Question : 84 of 90

Marks: +1, -0

C_0^2

C_1^2

C_2^2

C_n^2

, where n is an even integer is

Solution:

We have

(1+x^n)

C_0+C_1 x + C_2 x^2

+ ... +

C_n x^n

... (1)

\left(1-\frac{1}{x}\right)^n

C_0

\frac{C_1}{x}+\frac{C_2}{x^2}

+ ... +

(-1)^n \frac{C_n}{x^n}

... (2)

∴

C_0^2-C_1^2+C_2^2

... +

(-1)^nC_n^2

= co-eff. Of the term independent of x in Product of R.H.S. (1)and (2)

= co-eff. Of term independent of x in

(1+x)^n

\left(1-\frac{1}{x}\right)^n

= co-eff. Of

x^n

(-1)^n

(1-x^2)^n

\left\lceil \frac{n}{2} \right\rceil C_{n \text{ div } 2}

[∵ n is even]

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