Answer: Both (2n+1)Cn and (2n+1)C(n+1).
- A (2n+1)Cn, taken alone as the unique greatest coefficient
- B (2n+1)C(n+1), taken alone as the unique greatest coefficient
- C (2n)Cn, the central coefficient of the even-power expansion instead
- D Both (2n+1)Cn and (2n+1)C(n+1)
Correct answer: D. Both (2n+1)Cn and (2n+1)C(n+1)
Explanation: For (1+x)<sup>N</sup> with N=2n+1 (odd total power), the expansion has two equal middle terms T(n+1) and T(n+2) with coefficients (2n+1)Cn and (2n+1)C(n+1). These are equal since nCr=nC(n−r), so both are the greatest coefficients.
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