The coefficient of x⁵ in (1 + x)¹⁰ is:

Solution

Maths sir · 3 steps

Write the general term, match the power of x to 5, and the coefficient is just C(10, 5).

1. General term. In (1+x)¹⁰, the term with xʳ is Tᵣ₊₁ = C(10, r) 1¹⁰⁻ʳ xʳ = C(10, r)xʳ. The power of 1 is always 1, so the coefficient is only the binomial coefficient.
2. Match the power. You need x⁵, so set r = 5. The coefficient is C(10, 5), which sits in the 6th term T₆.

   C(10, 5) = (10 × 9 × 8 × 7 × 6)/(5 × 4 × 3 × 2 × 1) = 30240/120 = 252

3. So. The coefficient of x⁵ is 252, option (a).

The idea to keep. For the coefficient of xᵏ, set the power of x in the general term equal to k and solve for r. The term number is then r+1, not r.

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