The coefficient of x⁷ in (1 - x + 2x³)¹⁰ is ___________.

Solution

Maths sir · 3 steps

Three terms inside, so use the multinomial: list every way the powers of x can add up to 7.

1. General term. Pick 1 from a brackets, -x from b brackets and 2x³ from c brackets, with a + b + c = 10. That term is 10!/(a! b! c!)(-1)ᵇ 2ᶜ xᵇ⁺³ᶜ.
2. Find the cases. We need b + 3c = 7 with a = 10 - b - c ≥ 0. So c can be 0, 1 or 2, and then b = 7, 4, 1.

   (a, b, c)	10!/(a! b! c!)	(-1)ᵇ 2ᶜ	Contribution
   (3, 7, 0)	120	-1	-120
   (5, 4, 1)	1260	+2	2520
   (7, 1, 2)	360	-4	-1440

3. So. Coefficient of x⁷ is -120 + 2520 - 1440 = 960.

The common slip: 780. Forgetting the factor 2ᶜ from 2x³ gives -120 + 1260 - 360 = 780, not 960.

The idea to keep. For a trinomial raised to a power, solve the exponent equation for whole-number cases first. Then each case gives multinomial coefficient times the coefficients of the chosen terms, with their signs and powers.

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