<p>If number of terms in the expansion of <span>\((1 + 2x - 3y + 4z)^n\)</span> is 286, then find the coefficient of term containing <span>\(xyz\)</span>.</p>
Step-by-Step Solution
Key Concept: Use the multinomial expansion formula: the number of terms in (a₁ + a₂ + ... + aₖ)ⁿ is C(n+k-1, k-1). Here with 4 terms and 286 total terms, find n first, then use multinomial coefficients to find the coefficient of xyz.
<p><strong>Step 1: Find n using the number of terms formula</strong></p><p>For expansion of (a₁ + a₂ + a₃ + a₄)ⁿ, number of terms = C(n+3, 3) = 286</p><p>C(n+3, 3) = (n+3)(n+2)(n+1)/6 = 286</p><p>(n+3)(n+2)(n+1) = 1716</p><p>Testing: n=10: (13)(12)(11) = 1716 ✓</p><p>Therefore n = 10</p><p><strong>Step 2: Find coefficient of xyz in (1 + 2x - 3y + 4z)¹⁰</strong></p><p>We need the general term where powers sum to 10: a + b + c + d = 10, where the term is:</p><p>T = (10!)/(a!b!c!d!) · 1ᵃ · (2x)ᵇ · (-3y)ᶜ · (4z)ᵈ</p><p>For xyz term: b = 1, c = 1, d = 1, so a = 10 - 1 - 1 - 1 = 7</p><p><strong>Step 3: Calculate the multinomial coefficient and powers</strong></p><p>Coefficient = (10!)/(7!·1!·1!·1!) · 2¹ · (-3)¹ · 4¹</p><p>= (10 · 9 · 8)/(1) · 2 · (-3) · 4</p><p>= 720 · 2 · (-3) · 4</p><p>= 720 · (-24)</p><p>= -17280</p><p>∴ Answer: <strong>-17280</strong></p>
Correct Answer: -17280