Using factorials, show that \(\binom{n}{r-1}+\binom{n}{r}=\binom{n+1}{r}\).

Hence prove by mathematical induction that
\((a+x)^{n}=\binom{n}{0} a^{n}+\binom{n}{1} a^{n-1} x+\ldots+\binom{n}{r} a^{n-r} x^{r}+\ldots+\binom{n}{n} x^{n}\)
for every positive integer \(n\).