If the Coefficient of x³⁰ in the expansion of (1+1/x)⁶(1+x²)⁷(1-x³)⁸ ; x ≠ 0 is α, then |α| equals ___________.

Solution

Maths sir · 3 steps

Clear the 1/x first, then use symmetry so you hunt for x⁸ instead of x³⁶.

1. Clear the denominator. (1+1/x)⁶ = (1+x)⁶/x⁶. So the coefficient of x³⁰ in the given product is the coefficient of x³⁶ in P(x) = (1+x)⁶(1+x²)⁷(1-x³)⁸.
2. Use symmetry. P(x) has degree 6+14+24 = 44. Each factor reads the same backwards: (1-x³)⁸ reversed is (x³-1)⁸, which is the same thing because the power 8 is even. So the coefficient of x³⁶ equals the coefficient of x⁴⁴⁻³⁶ = x⁸.

   C(6, a) xᵃ·C(7, b) x²ᵇ·(-1)ᶜC(8, c) x³ᶜ, a+2b+3c = 8

   c	(-1)ᶜC(8, c)	(a,b) with a+2b = 8-3c	Contribution
   0	1	(0,4),(2,3),(4,2),(6,1): 35+525+315+7 = 882	882
   1	-8	(1,2),(3,1),(5,0): 126+140+6 = 272	-2176
   2	28	(0,1),(2,0): 7+15 = 22	616

3. So. α = 882 - 2176 + 616 = -678. That gives |α| = 678.

The common slip: 3674. Dropping the (-1)ᶜ sign from (1-x³)⁸ makes every case positive and gives 882 + 2176 + 616 = 3674.

The idea to keep. When a product of binomials is palindromic, the coefficient of xᵏ equals that of xⁿ⁻ᵏ, where n is the total degree. Always pick the smaller power, and carry the (-1)ᶜ sign from minus terms.

Answer from the NTA final answer key. Solution by PrepWiser.