Mathematical induction on binomial coefficients

Question

I need to prove the following statement (Pascals Identity) on binomial coefficients using mathematical induction only

$$\binom{n}{r} = \binom{n-1}{r}+\binom{n-1}{r-1}$$

My doubt is

Whether I need to prove the following two statements or only the first one?

$$\binom{n+1}{r} = \binom{n}{r}+\binom{n}{r-1}$$

$$\binom{n}{r+1} = \binom{n-1}{r+1}+\binom{n-1}{r}$$

Answer 1

For Mathematical induction, you prove $$\binom{n}{r} = \binom{n-1}{r}+\binom{n-1}{r-1}$$

for $n=1$.

Then assume that $$\binom{n}{r} = \binom{n-1}{r}+\binom{n-1}{r-1}$$ is true for $n=k$, and prove it for $n=k+1$

You do not need to change $r$ to $r+1$

This problem may be easier to solve without Mathematical Induction, but it is your choice.