Question

Which set of coefficients of the terms in the expansion of the binomial (x + y)3 is correct

Answer 1

By the binomial theorem,
\[(ax+by)^n=\sum_{k=0}^n\binom nk (ax)^{n-k}(by)^k=\sum_{k=0}^n\binom nk a^{n-k}b^kx^{n-k}y^k\]which means the set of coefficients for any fixed positive integer \(n\) must be
\[\left\{\binom n0a^n,\binom n1a^{n-1}b,\binom n2a^{n-2}b^2,\ldots,\binom n{n-1}ab^{n-1},\binom nnb^n\right\}\]Note that for \(0\le k\le n\), you have \(\dbinom nk=\dbinom n{n-k}\), so there will be repeats.

In your case, \(a=1\), \(b=1\), and \(n=3\).