TITLE: Binomial Theorem Identities
QUESTION [9 upvotes]: What's the actual difference between these two formulas (they're both in the chapter regarding binomial theorem).  They're from two different textbooks :
$${n\choose k}+{n\choose k+1}={n+1\choose k+1}$$
and 
$${n-1\choose k}+{n-1\choose k-1}={n\choose k}$$
I'll be appreciated if someone explain it to me either combinatorially or algebraically.  Thanks!

REPLY [2 votes]: Combinatorial proof of Pascal identity
Denote by $$C^k(I_n)=\{A:A\subset I_n,|A|=k\},|C^k(I_n)|=\binom{n}{k}$$ the set of all k-subsets of set $I_n=\{0,1,2,...,n-2,n-1\}$ and then define a set 
$$C'=\{A:A=C_1\cup\{n-1\},C_1\in C^{k-1}(I_{n-1})\}$$
now is clear that
$$|C'|=|C^{k-1}(I_{n-1})|=\binom{n-1}{k-1}$$
because
$$C^k(I_n)=C'\cup C^k(I_{n-1})$$
$$C'\cap C^k(I_{n-1})=\emptyset$$
follow that
$$|C^k(I_n)|=|C'|+|C^k(I_{n-1})|$$
or
$$\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}$$