The coefficient of x⁷ in the expression
(1 + x)¹⁰ + x(1 + x)⁹ + x²(1 + x)⁸ + ......+ x¹⁰ is:

Solution

Maths sir · 4 steps

Each term is the previous one times x/(1+x), so sum it as a GP and the whole thing collapses.

1. Spot the GP. The terms are xᵏ(1+x)¹⁰⁻ᵏ for k = 0 to 10, which is 11 terms. Each term is the one before it multiplied by r = x/(1+x), starting from a = (1+x)¹⁰.

   S = (1+x)¹⁰·(1-(x/(1+x))¹¹)/(1-x/(1+x)) = (1+x)¹¹ - x¹¹

2. Read off the coefficient. The term x¹¹ does not touch x⁷. So the coefficient of x⁷ is just C(11, 7) = C(11, 4) = (11 × 10 × 9 × 8)/24 = 330.
3. Quick check. Term by term, the contributions are C(10, 3) + C(9, 3) + … + C(3, 3) = 120 + 84 + 56 + 35 + 20 + 10 + 4 + 1 = 330. The two methods agree.
4. So. The coefficient of x⁷ is 330, option (b).

The idea to keep. When the powers of x go up while the powers of (1+x) come down by one each time, you are looking at a GP. Sum it first and you get one clean binomial instead of eight coefficients.

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