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In the expansion of (1 + x + x³ + x⁴)¹⁰, find the coefficient of x⁴.

1. 40C4
2. 10C4
3. 210
4. 310

Correct answer: 210

Solution

(1+x+x³+x⁴)¹⁰ = (1+x)¹⁰*(1+x³)¹⁰. Coefficient of x⁴ = C(10,4)*1 + C(10,1)*C(10,1) = 210 + 100 = 310. Wait: C(10,4)=210 (from x⁴ in (1+x)¹⁰ times x⁰) plus C(10,1)*C(10,1)=10*10=100 (from x¹ times x³). Total = 310.

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