The coefficient of x⁴ in the expansion of (1+x+x²+x³)^(11) is what?

i) Factorizing 1 + x + x² + x³ = (1 + x²)*(1 + x)

ii) So, (1 + x + x² + x³)¹¹ = (1 + x²)¹¹*(1 + x)¹¹
Applying binomial expansion, the above =
= {1 + C1*(x²) + C2*(x⁴) + C3*(x⁶) + ......}*{1 + C1*x + C2*(x²) + C3*(x³) + C4*(x⁴) + .....}

iii) Multiplying both the above,
Coefficient of x⁴ is = 1*C4 + (C1)*(C2) + (C2)*(1) = 330 + 11*55 + 55 = 330 + 605 + 55 = 990
[C4 = C(11, 4) = 11*10*9*8/(1*2*3*4) = 330; C1 = 11; C2 = 11*10/1*2 = 55]

(x³+x²+x+1)¹¹ = ((x²+1)(x+1))¹¹ = (x²+1)¹¹(x=1)¹¹
The expansion of (x²+1)¹¹ contains even powers of x.
x⁴ = x⁴·1 = x²x²

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