In the expansion of (1 + x) n + 2 , the coefficient of x n is (n+2)C n. Using this, find the coefficient of x² in (1 + x)⁵ + (1 + x)⁶ + ... + (1 + x)¹⁰.

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In the expansion of (1 + x)n + 2, the coefficient of xn is (n+2)C n. Using this, find the coefficient of x² in (1 + x)⁵ + (1 + x)⁶ + ... + (1 + x)¹⁰.

Answer: 155.

A 145
B 155
C 165
D 175

Correct answer: B. 155

Explanation: Coefficient of x² in (1+x)k is kC2. Sum for k=5 to 10: 5C2+6C2+7C2+8C2+9C2+10C2 = 10+15+21+28+36+45 = 155. (By the hockey-stick identity, this also equals 11C3 - 5C3 = 165-10 = 155.)

Row n of Pascal's triangle gives the coefficients nC₀, nC₁, ..., nCn for the expansion of (a+b)ⁿ; each number is the sum of the two numbers diagonally above it, a direct visual proof of the identity nCr = (n-1)C(r-1) + (n-1)Cr.

Concept context

Expansion of (a+b) n , general term, middle term, binomial coefficients, and greatest term

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In the expansion of (1 + x)<sup>n + 2</sup>, the coefficient of x<sup>n</sup> is (n+2)C n. Using this, find the coefficient of x² in (1 + x)⁵ + (1 + x)⁶ + ... + (1 + x)¹⁰.

Concept context

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