Is this a valid operation on binomial coefficients?

Question

I'm trying to prove all binomial coefficients are natural numbers by induction. From another part of the question it is given that ${n \choose k} = {n-1 \choose k-1} + {n-1 \choose k}$

Can I add 1 to n and say ${n + 1 \choose k} = {n \choose k-1} + {n \choose k}$

I should mention that $n$ is a natural number.

Is that a valid operation? I only ask this because for induction I want to prove if ${n \choose k}$ is a natural number then so is ${n+1 \choose k}$.

Answer 1

That depends on what you are given. If you are given that formula for all $n\geqslant 1$, then yes, in that case you are just basically plugging in $n=t+1$ and then renaming $t$ to $n$. If you are given that equality for a fixed $n$ (which is more likely in a proof by induction), then no, you can't perform such operations. Basically, just check whether $n$ is a free variable or some fixed number beforehand.

Answer 2

Yes you can chang $n$ to $n+1$ but that will alter the range of $k$ in the formula. In your proof be careful to address that issue.

Answer 3

You have to define precisely what you want to prove by induction on $n$, which is

for all $k \geqslant 0$, $\binom{n}{k}$ is an integer.

In order to do this, you first need to prove the result for $n = 0$, that is

for all $k \geqslant 0$, $\binom{0}{k}$ is an integer

This is of course very easy. Then you can apply the induction hypothesis and your formula to conclude.