A possible proof of Fermat's Little Theorem

Question

By the formula $$(a+b)^n=\sum_{k=0}^n \binom{n}{k}a^kb^{n-k}$$ we know that $(a+b)^p\equiv a^p+b^p \pmod{p}$,

Is there a proof of Fermat's Little Theorem based on this fact?

Answer 1

As noted in the comment by Lord Shark, using that fact, for the induction step, assuming as hypotesis $a^p\equiv a \mod p$ we have

$$(a+1)^p\equiv a^p+1\equiv a+1 \mod p$$