Proof:
We have to show that
[tex]\binom{n}{r-1}+\binom{n}{r}=\binom{n+1}{r}[/tex]Solving the LHS of the above relation we have[tex]\\\\\frac{n!}{(n-(r-1))!\cdot (r-1)!}+\frac{n!}{(n-r)!\cdot r!}\\\\\frac{n!}{(n-r+1)!\cdot (r-1)!}+\frac{n!}{(n-r)!\cdot r!}\\\\\frac{n!}{(n-r+1)!\cdot (r-1)!}\times \frac{r}{r}+\frac{n!}{(n-r)!\cdot r!}\times \frac{n-r+1}{n-r+1}\\\\\frac{rn!}{(n-r+1)!\cdot r!}+\frac{(n-r+1)\cdot n!}{(n-r+1)!\cdot r!}\\\\\frac{n!}{(n-r+1)!\cdot r!}\cdot (r+n-r+1)\\\\\frac{(n+1)!}{(n-r+1)!\times r!}=\binom{n+1}{r}[/tex]
Hence proved
