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Which of the following binomial coefficient identities are TRUE?

1. C(n,0)*C(n,1) + C(n,1)*C(n,2) +... + C(n,n-1)*C(n,n) = C(2n, n+1)
2. [C(n,0)]² + [C(n,1)]² + [C(n,2)]² +... + [C(n,n)]² = C(2n, n)
3. C(11,0)*C(11,11) - C(11,1)*C(11,10) + C(11,2)*C(11,9) -... + (-1)¹¹ * C(11,11)*C(11,0) = C(22,11)
4. C(n,0) + 2*C(n,1)/2 + 3*C(n,2)/3 +... + (n+1)*C(n,n)/(n+1) = (3^(n+1) - 1)/(n+1)

Correct answer: [C(n,0)]² + [C(n,1)]² + [C(n,2)]² +... + [C(n,n)]² = C(2n, n)

Solution

A: By Vandermonde's identity, sum_(r=0)ⁿ⁻¹ C(n,r)*C(n,r+1) = C(2n, n-1) = C(2n, n+1). TRUE. B: sum [C(n,r)]² = C(2n,n). TRUE (standard result). C: sum_(r=0)¹¹ (-1)^r [C(11,r)]² = coefficient of x¹¹ in (1-x)¹¹*(1+x)¹¹ = (1-x²)¹¹. All powers of (1-x²)¹¹ are even, so coefficient of x¹¹ is 0, not C(22,11). FALSE. D: The sum simplifies to sum_(r=0)ⁿ C(n,r) = 2ⁿ (after cancellation), not (3^(n+1)-1)/(n+1). FALSE. Correct: A and B.

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