[Solved] If C0, C1, C2, _ _ _ _ _, Cn Are The Coefficients In The Exp

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Concept:

(1 + x)n = C0 + x. C1 + x2. C2 + ...... + xnCn

Calculation:

Let S = C1 + C2 + C3 + _ _ _ _ _ + Cn

⇒ S = nC1 + nC2 + nC3 + _ _ _ _ _ + nCn

⇒ S = 1 + nC1 + nC2 + nC3 + _ _ _ _ _ + nCn - 1

⇒ S = nC0 + nC1 + nC2 + nC3 + _ _ _ _ _ + nCn - 1 ----(1)

As we know, (1 + x)n = C0 + x. C1 + x2. C2 + ...... + x nCn

Put x = 1, we get

⇒ (1 + x)n = nC0 + nC1 + nC2 + nC3 + .... + nCn = 2n -----(2)

From equation (1) and (2), we get

S = nC0 + nC1 + nC2 + nC3 + _ _ _ _ _ + nCn - 1

∴ The value of C1 + C2 + C3 + _ _ _ _ _ + Cn is 2n - 1.

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