Answer: (2n)Cn.
- A 2ⁿ
- B (2n)Cn
- C 2nCn
- D n!
Correct answer: B. (2n)Cn
Explanation: Vandermonde identity: Σ(r=0 to n)(nCr)²=coeff of xⁿ in (1+x)<sup>n</sup>·(1+x)<sup>n</sup>=(1+x)<sup>2n</sup>, which equals (2n)Cn. Since nCr=nC(n−r), this is equivalent to coeff of xⁿ in (1+x)<sup>2n</sup>. Answer: (2n)Cn.
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