Doubt about $r$th Term in Binomial theorem

Question

When asked about $10$th term in expansion of $(a+b)^{15}$ we have

$$T_{10}=\binom{15}{10}a^5b^{10}$$

But we can also write the binomial as $(b+a)^{15}$ and say $10$th term as

$$T_{10}=\binom{15}{10}b^5a^{10}$$

which is correct now?

Answer 1

Keeping the given order, with $a$ as first, I would select the first one that is $T_{10}=\binom{15}{10}a^5b^{10}$.

But you are right the question posed in that way is not much clear and ity should be more correct ask for example what is the term containing $a^5$.

Answer 2

If you're asked to write the $10$-th coefficient of the expansion $(a+b)^{15}$ it is $$T_{10}={15\choose 10}a^5b^{10}$$ because you were specifically asked to take the expansion of $(a+b)^n$ and not $(b+a)^n$. In any case the $10$-th coefficient of the expansion of $(a+b)^{15}$ is the $5$-th coefficient of $(b+a)^{15}$. The entire expansion of both expressions are the same, just rearranged in a different way. As an example let's expand $(a+b)^5$ and $(b+a)^5$: $$(a+b)^5 = a^5+\color{red}{{5\choose 1}}a^4b+\color{blue}{{5\choose 2}}a^3b^2+\color{orange}{{5\choose 3}}a^2b^3+\color{green}{{5\choose 4}}ab^4+b^5 \\ (b+a)^5 = b^5+\color{green}{{5\choose 1}}b^4a+\color{orange}{{5\choose 2}}b^3a^2+\color{blue}{{5\choose 3}}b^2a^3+\color{red}{{5\choose 4}}ba^4+a^5$$ where same color represents same value of the coefficients by means of $${n\choose k} = {n\choose {n-k}}$$