Proof Using Multinomial Coefficients Verification

Question

Use multinomial coefficients to prove that, for all positive integers n,

$6^n=\binom{n}{0}5^n+\binom{n}{1}5^{n-1}+\binom{n}{2}5^{n-2}+\cdots +\binom{n}{n}5^0$

Provide a clear proof and all working, justifying and explaining all steps taken.

I have posted this question before, but I had included a different proof to the one I'm including now. Hopefully, this will be a little easier to interpret.

The proof I've come up with is below:

Is this proof okay? I'm open to any suggestions! :)

Answer 1

Here's a concise way to write it. By the Binomial Theorem, $$ \left(1+x\right)^{n} =\sum_{k=0}^{n}\binom{n}{k}1^{k}x^{n-k} =\sum_{k=0}^{n}\binom{n}{k}x^{n-k}. $$ Taking $x=5$ gives you the special case you mention.