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

Question

holsteremission · Answer

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 $n=3$.