If C₀, C₁, C₂, _ _ _ _ _, C_n are the coefficients in the expansion of (1 + x)ⁿ, then what is the value of C₁ + C₂ + C₃ + _ _ _ _ _ + C_n?

Concept:

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

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 nC_n

Put x = 1, we get

⇒ (1 + x)n =  nC0 + nC1 + nC2 + nC3 + .... + nC_n = 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.