What is the story behind ${n+1 \choose k} = {n \choose k} + {n \choose k-1}$?
424
Proving Pascal's Rule : ${{n} \choose {r}}={{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$
46132
Combinatorial interpretation for the identity $\sum\limits_i\binom{m}{i}\binom{n}{j-i}=\binom{m+n}{j}$?
68080
Evaluation of $\sum_{k=0}^n{n\choose k}^2u^k$
136844
Mathematical induction question: why can we "assume $P(k)$ holds"?
34452
In how many ways can you choose $k$ numbers out of $\{1,2,3,\dots,n\}$ so none of them is consecutive?
133905
Intuitive proof of $\sum_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k} = n^n$
119759
Help ending a proof using binomial theorem: $\sum_{k=0}^n {n \choose k}\cdot\frac{(-1)^k}{k+1} = \frac{1}{n+1}$
8476
Proof of the Hockey-Stick Identity: $\sum\limits_{t=0}^n \binom tk = \binom{n+1}{k+1}$
89576
Proof of a binomial identity $\sum_{k=0}^n {n \choose k}^{\!2} = {2n \choose n}.$
142666
Algebric proof for the identity $n(n-1)2^{n-2}=\sum_{k=1}^n {k(k-1) {n \choose k}}$
67927
Combinatorial argument for the identity $k\binom{n}{k} = n\binom{n-1}{k-1}$
925
Prove that $F_n={n-1 \choose 0 }+{n-2 \choose 1 }+{n-3 \choose 2 }+\ldots$ where $F(n)$ is the $n$-th fibonacci number
124116
What does the notation $\binom{n}{i}$ mean?
136037
${n\choose 2} - {k\choose 2} = {n-k\choose 2} +k(n-k)$ Marked box or unmarked box
24588
Proving the combinatorial identity ${n \choose k} = {n-2\choose k-2} + 2{n-2\choose k-1} + {n-2\choose k}$
121279
Proving an Combination formula $ \binom{n}{k} = \binom{n-1}{k}+\binom{n-1}{k-1}$
73102
Given the factors of $N$, is there a method for computing the factors of $N-1$ or $N+1$?