If 20C1 + (22 )20C2 + (32)20C3 +…..+ (202) 20C20 = A(2β)¸then the ordered pair (A, β) is equal to :

Engineering

Mathematics

Properties of Binomial Coefficients

Question

If ²⁰C₁ + (2² )²⁰C₂ + (3²)²⁰C₃ +…..+ (20²) ²⁰C₂₀ = A(2^β)¸then the ordered pair (A, β) is equal to :

(420, 19)

(380, 19)

(380, 18)

(420, 18)

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(1 + x)²⁰ = ²⁰C₀ + ²⁰C₁x + ²⁰C₂x² +….+ ²⁰C₂₀x²⁰ …..(i)

different eqⁿ (i) w.r.t x

20(1 + x)¹⁹ = ²⁰C₁.1 + 2.²⁰C₂x +….+ 20²⁰C₂₀x¹⁹ ….(ii)

Multiply eqⁿ (ii) by x

20x(1 + x)¹⁹ = ²⁰C₁.x + 2.²⁰C₂x² +..…+ 20²⁰C₂₀x²⁰ .…(iii)

diff. eqⁿ (iii) w.r.t. x

20[1 + x]¹⁹ + 19x(1 + x)¹⁸] = 1.¹⁰C₁ + 2².²⁰C₂x¹⁹ …..(iv)

put x = 1 in eqⁿ (iv)

20(2¹⁹ + 19.2¹⁸) = 1².²⁰C₁ + 2².²⁰C₂ +……+ (20²)²⁰C₂₀

= 20 × 2¹⁸(2 + 19) = 20 × 21 × 2¹⁸ = 420 × 2¹⁸

A = 420, β = 18