The coefficient of x¹⁸ in the expansion of (x⁴-1/x³)¹⁵ is __________.

Solution

Maths sir · 4 steps

Write the general term, collect all the powers of x into one exponent, and set it equal to 18.

1. General term. In (a+b)¹⁵ the general term is Tᵣ₊₁ = C(15, r) a¹⁵⁻ʳ bʳ. Here a = x⁴ and b = -1/x³ = -x⁻³.

   Tᵣ₊₁ = C(15, r)(x⁴)¹⁵⁻ʳ(-x⁻³)ʳ = (-1)ʳC(15, r) x⁶⁰⁻⁷ʳ

2. Match the power. Set 60 - 7r = 18. Then 7r = 42, so r = 6. This is the term T₇.
3. Find the coefficient. The coefficient is (-1)⁶C(15, 6). Since 6 is even the sign is positive, and C(15, 6) = (15 × 14 × 13 × 12 × 11 × 10)/720 = 5005.
4. So. The coefficient of x¹⁸ is 5005.

The common slip: 6435. Using the term number 7 as the index and writing C(15, 7) gives 6435. The term Tᵣ₊₁ with r = 6 uses C(15, 6).

The idea to keep. For any term in a binomial expansion, write Tᵣ₊₁, turn every power of x into one exponent in r, and solve. Keep the (-1)ʳ so the sign comes out right.

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