What is C(n, 1) + C(n, 2) + _ _ _ _ _ + C(n, n) equal to

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NDA (Held On: 18 Apr 2021) Maths Previous Year paper

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Answer 1

Concept:

(1 + x)n = nC0 × 1(n-0) × x 0+ nC1 × 1(n-1) × x 1 + nC2 × 1(n-2) × x2 + …. + nCn × 1(n-n) × xn

n^th term of the G.P. is an = arⁿ⁻¹

Sum of n terms = s = ; where r >1

Sum of n terms = s = ; where r <1

Calculation:

C(n, 1) + C(n, 2) + _ _ _ _ _ + C(n, n)

⇒ ⁿC₁ + ⁿC₂ + ... + ⁿC_n

⇒ ⁿC₀ + nC1 + nC2 + ... + nCn - ⁿC₀

⇒ (1 + 1)ⁿ - ⁿC_o

⇒ 2ⁿ - 1 = = 1 ×

Comparing it with a G.P sum = a × , we get a = 1 and r = 2

∴ 2n - 1 = 1 + 2 + 2² + ... +2^n-1 which will give us n terms in total.