Let a and b be the coefficient of x3 in (1 + x + 2x2 + 3x3)3 and (1 + x + 2x2 + 3x3 + 4x4)3, respectively then

Engineering

Mathematics

Introduction to Binomial Theorem

Question

Let a and b be the coefficient of x³ in (1 + x + 2x² + 3x³)³ and (1 + x + 2x² + 3x³ + 4x⁴)³, respectively then

a = b

a < b

a + b = 22

a > b

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(1 + z)³ where z = x(1 + 2x + 3x²)
1 + ³C₁z + ³C₂z² + ³C₃z³
coefficient of x³ in (1 + z)³
³C₁(3) + ³C₂ (4) + ³C₃ (1) = 22
⇒   a = 22
now again (1 + y)³
where y = x (1 + 2x + 3x² + 4x³)
(1 + y)³ = 1 + ³C₁y + ³C₂y² + ³C₃y³
$∴$    coefficient of x³ is
³C₁(3) + ³C₂ (4) + ³C₃ (1)
= 9 + 12 + 1 = 22
⇒   b = 22
Hence a = b   ⇒   a + b = 44

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