# A problem in proving with induction

According to asked question in this post.

Suppose $$T(n,k)=T(n-1,k-1)+T(n-1,k)+1$$, now let $$C(n,k)=T(n,k)+1$$.

As a result $$C(n,k)=C(n-1,k-1)+C(n-1,k)$$.

I want to prove $$C(n,k)=2\binom{n}{k}$$, now on that post mentioned that use induction to prove equality such that induction on $$k+n$$. My problem is when i use induction on $$k+n=m$$, And define $$P(m)$$ I can't accurately prove. For example: when i try to prove $$P(2)$$, i set $$k=0,n=2$$ or $$k=1,n=1$$ and for $$P(2)$$ we have two cases. Anyone can help me to prove it ?

$$P(m)$$ states that whenever $$0≤k≤n$$ and $$k+n=m$$ then $$C(n,k)=2\binom{n}{k}$$