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Show That C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = 128 - Mathematics and Statistics

Sum

Show That C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = 128

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Solution

We know that,

C0 + C1 + C2 + ... + Cn–1 + Cn = 2n

Put n = 8, we get,

C0 + C1 + C2 + C3 + ... + C7 + C8 = 28 = 256 ... (1)

We know that, the sum of even coefficients is equal to the sum of odd coefficients.

∴ C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = k ... (2)

From (1),

(C0 + C2 + C4 + C6 + C8) + (C1 + C3 + C5 + C7) = 256

∴ k + k = 256    ...[By (2)]

∴ 2k = 256

∴ k = 128

∴ C0 + C2 + C4 + C6 + C8 = C1 + C3 + C5 + C7 = 128.

Concept: Binomial Coefficients
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 11th Standard Maharashtra State Board
Chapter 4 Methods of Induction and Binomial Theorem
Exercise 4.5 | Q 5 | Page 84
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